Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions aid your preparation & make you hold a good grip on the easy to the complex level of topics covered in the chapter. You can easily & instantly get help in your homework and assignment questions by referring to this Ch 4 Polynomial Functions BIM Algebra 2 Textbook Answers. Make the most out of these Big Ideas Math book algebra 2 solution key of ch 4 polynomial functions and score maximum marks in all the examinations.

Big Ideas Math Book Algebra 2 Answer Key Chapter 4 Polynomial Functions

Enhancing your subject knowledge is even more important than scoring good marks. So, preparaing the ch 4 concepts from the subject experts provided Big Ideas Math Textbook Algebra 2 Answers Chapter 4 Polynomial Functions is even more essential. Both students and teachers can learn the math concepts in a simple and fun learning way by taking the help of BIM Algebra Textbook Answers. So, click on the links available below and practice answers covered in Questions from Chapter Tests, Review Tests, Cumulative Practice, Cumulative Assessments, Exercise Questions, etc.

Polynomial Functions Maintaining Mathematical Proficiency

Simplify the expression.

Question 1.
6x − 4x

Question 2.
12m − m − 7m + 3

Question 3.
3(y + 2) − 4y

Question 4.
9x − 4(2x − 1)

Question 5.
−(z + 2) − 2(1 − z)

Question 6.
−x2 + 5x + x2

Find the volume of the solid.

Question 7.
cube with side length 4 inches

Question 8.
sphere with radius 2 feet

Question 9.
rectangular prism with length 4 feet, width 2 feet, and height 6 feet

Question 10.
right cylinder with radius 3 centimeters and height 5 centimeters

Question 11.
ABSTRACT REASONING Does doubling the volume of a cube have the same effect on the side length? Explain your reasoning.

Polynomial Functions Mathematical Practices

Monitoring Progress

Use a graphing calculator to determine whether the function is continuous. Explain your reasoning.

Question 1.
f(x) = \(\frac{x^{2}-x}{x}\)

Question 2.
f(x) = x3 − 3

Question 3.
f(x) = \(\sqrt{x^{2}+1}\)

Question 4.
f(x) = | x + 2 |

Question 5.
f(x) = \(\frac{1}{x}\)

Question 6.
f(x) = \(\frac{1}{\sqrt{x^{2}-1}}\)

Question 7.
f(x) = x

Question 8.
f(x) = 2x − 3

Question 9.
f(x) = \(\frac{x}{x}\)

Lesson 4.1 Graphing Polynomial Functions

Essential Question
What are some common characteristics of the graphs of cubic and quartic polynomial functions?
A polynomial function of the form
f(x) = anxn + an – 1xn– 1 +. . .+ a1x + a0
where an ≠ 0, is cubic when n = 3 and quartic when n = 4.

EXPLORATION 1
Identifying Graphs of Polynomial Functions
Work with a partner.
Match each polynomial function with its graph. Explain your reasoning. Use a graphing calculator to verify your answers.
a. f(x) = x3 − x
b. f(x) = −x3 + x
c. f(x) = −x4 + 1
d. f(x) = x4
e. f(x) = x3
f. f(x) = x4 − x2

EXPLORATION 2
Identifying x-Intercepts of Polynomial Graphs
Work with a partner.
Each of the polynomial graphs in Exploration 1 has x-intercept(s) of −1, 0, or 1. Identify the x-intercept(s) of each graph. Explain how you can verify your answers.

Communicate Your Answer

Question 3.
What are some common characteristics of the graphs of cubic and quartic polynomial functions?

Question 4.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 1
Determine whether each statement is true or false. Justify your answer.
a. When the graph of a cubic polynomial function rises to the left, it falls to the right.
b. When the graph of a quartic polynomial function falls to the left, it rises to the right.

4.1 Lesson

Monitoring Progress

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

Question 1.
f(x) = 7 − 1.6x2 − 5x

Question 2.
p(x) =x+ 2x-2 + 9.5

Question 3.
q(x) = x3 − 6x + 3x4

Evaluate the function for the given value of x.

Question 4.
f(x) = −x3 + 3x2 + 9; x = 4

Question 5.
f(x) = 3x5 − x4 − 6x + 10; x = −2

Question 6.
Describe the end behavior of the graph of f(x) = 0.25x3 − x2 − 1.

Graph the polynomial Function

Question 7.
f(x) = x4 + x2 − 3

Question 8.
f(x) = 4 − x3

Question 9.
f(x) = x3 − x2 + x − 1

Question 10.
Sketch a graph of the polynomial function f having these characteristics.

  • f is decreasing when x < −1.5 and x > 2.5; f is increasing when −1.5 < x < 2.5.
  • f(x) > 0 when x < −3 and 1 < x < 4; f(x) < 0 when −3 < x < 1 and x > 4.

Use the graph to describe the degree and leading coefficient of f.

Question 11.
WHAT IF? Repeat Example 6 using the alternative model for electric vehicles of
V(t) = −0.0290900t4 + 0.791260t3 − 7.96583t2 + 36.5561t − 12.025.

Graphing Polynomial Functions 4.1 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING Explain what is meant by the end behavior of a polynomial function.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 1

Question 2.
WHICH ONE DOESN’T BELONG? Which function does not belong with the other three? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 2

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

Question 3.
f(x) = −3x + 5x3 − 6x2 + 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 3

Question 4.
p(x) = \(\frac{1}{2}\)x2 + 3x − 4x3 + 6x4 − 1
Answer:

Question 5.
f(x) = 9x4 + 8x3 − 6x-2 + 2x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 5

Question 6.
g(x) = \(\sqrt{3}\) − 12x + 13x2
Answer:

Question 7.
h(x) = \(\frac{5}{3}\)x2 − \(\sqrt{7}\)x4 + 8x3 − \(\frac{1}{2}\) + x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 7

Question 8.
h(x) = 3x4 + 2x − \(\frac{5}{x}\) + 9x3 − 7
Answer:

ERROR ANALYSIS In Exercises 9 and 10, describe and correct the error in analyzing the function.

Question 9.
f(x) = 8x3 − 7x4 − 9x − 3x2 + 11
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 9

Question 10.
f(x) = 2x4 + 4x – 9\(\sqrt{x}\) + 3x2 – 8
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4
Answer:

In Exercises 11–16, evaluate the function for the given value of x.

Question 11.
h(x) = −3x4 + 2x3 − 12x − 6; x = −2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 11

Question 12.
f(x) = 7x4 − 10x2 + 14x − 26; x = −7
Answer:

Question 13.
g(x) = x6 − 64x4 + x2 − 7x − 51; x = 8
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 13

Question 14.
g(x) = −x3 + 3x2 + 5x + 1; x = −12
Answer:

Question 15.
p(x) = 2x3 + 4x2 + 6x + 7; x = \(\frac{1}{2}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 15

Question 16.
h(x) = 5x3 − 3x2 + 2x + 4; x = −\(\frac{1}{3}\)
Answer:

In Exercises 17–20, describe the end behavior of the graph of the function.

Question 17.
h(x) = −5x4 + 7x3 − 6x2 + 9x + 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 17

Question 18.
g(x) = 7x7 + 12x5 − 6x3 − 2x − 18
Answer:

Question 19.
f(x) = −2x4 + 12x8 + 17 + 15x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 19

Question 20.
f(x) = 11 − 18x2 − 5x5 − 12x4 − 2x
Answer:

In Exercises 21 and 22, describe the degree and leading coefficient of the polynomial function using the graph.

Question 21.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 21

Question 22.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 6
Answer:

Question 23.
USING STRUCTURE Determine whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.
f(x) = 5x3x + \(\frac{5}{2}\)x3 − 9x4 + \(\sqrt{2}\)x2 + 4x − 1 − x-5x5 − 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 23

Question 24.
WRITING Let f(x) = 13. State the degree, type, and leading coefficient. Describe the end behavior of the function. Explain your reasoning.
Answer:

In Exercises 25–32, graph the polynomial function.

Question 25.
p(x) = 3 − x4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 25

Question 26.
g(x) = x3 + x + 3
Answer:

Question 27.
f(x) = 4x − 9 − x3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 27

Question 28.
p(x) = x5 − 3x3 + 2
Answer:

Question 29.
h(x) = x4 − 2x3 + 3x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 29

Question 30.
h(x) = 5 + 3x2 − x4
Answer:

Question 31.
g(x) = x5 − 3x4 + 2x − 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 31

Question 32.
p(x) = x6 − 2x5 − 2x3 + x + 5
Answer:

ANALYZING RELATIONSHIPS In Exercises 33–36, describe the x-values for which
(a) f is increasing or decreasing,
(b) f(x) > 0, and
(c) f(x) < 0.

Answer:

Question 33.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 33

Question 34.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 8
Answer:

Question 35.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 35

Question 36.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 10
Answer:

In Exercises 37–40, sketch a graph of the polynomial function f having the given characteristics. Use the graph to describe the degree and leading coefficient of the function f.

Question 37.

  • f is increasing when x > 0.5; f is decreasing when x < 0.5.
  • f(x) > 0 when x < −2 and x > 3; f(x) < 0 when −2 < x < 3.

Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 37

Question 38.

  • f is increasing when −2 < x < 3;f is decreasing when x < −2 and x > 3.
  • f(x) > 0 when x < −4 and 1 < x < 5; f(x) < 0 when −4 < x < 1 and x > 5.

Answer:
Question 39.

  • f is increasing when −2 < x < 0 and x > 2;f is decreasing when x < −2 and 0 < x < 2.
  • f(x) > 0 when x < −3, −1 < x < 1, and x > 3; f(x) < 0 when −3 < x < −1 and 1 < x < 3.

Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 39

Question 40.

  • f is increasing when x < −1 and x > 1;f is decreasing when −1 < x < 1.
  • f(x) > 0 when −1.5 < x < 0 and x > 1.5; f(x) < 0 when x < −1.5 and 0 < x < 1.5.

Answer:
Question 41.
MODELING WITH MATHEMATICS From 1980 to 2007 the number of drive-in theaters in the United States can be modeled by the function
d(t) = −0.141t3 + 9.64t2 − 232.5t + 2421
where d(t) is the number of open theaters and t is the number of years after 1980.
a. Use a graphing calculator to graph the function for the interval 0 ≤ t ≤ 27. Describe the behavior of the graph on this interval.
b. What is the average rate of change in the number of drive-in movie theaters from 1980 to 1995 and from 1995 to 2007? Interpret the average rates of change.
c. Do you think this model can be used for years before 1980 or after 2007? Explain.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 11
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 41.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 41.2

Question 42.
PROBLEM SOLVING The weight of an ideal round-cut diamond can be modeled by
w = 0.00583d3 − 0.0125d2 + 0.022d − 0.01
where w is the weight of the diamond (in carats) and d is the diameter (in millimeters). According to the model, what is the weight of a diamond with a diameter of 12 millimeters?
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 12
Answer:

Question 43.
ABSTRACT REASONING Suppose f(x) → ∞ as x → −∞ and f(x) →−∞ as x → ∞. Describe the end behavior of g(x) = −f(x). Justify your answer.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 43

Question 44.
THOUGHT PROVOKING Write an even degree polynomial function such that the end behavior of f is given by f(x) → −∞ as x → −∞ and f(x) → −∞ as x → ∞. Justify your answer by drawing the graph of your function.
Answer:

Question 45.
USING TOOLS When using a graphing calculator to graph a polynomial function, explain how you know when the viewing window is appropriate.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 45

Question 46.
MAKING AN ARGUMENT Your friend uses the table to speculate that the function f is an even degree polynomial and the function g is an odd degree polynomial. Is your friend correct? Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 13
Answer:

Question 47.
DRAWING CONCLUSIONS The graph of a function is symmetric with respect to the y-axis if for each point (a, b) on the graph, (−a, b) is also a point on the graph. The graph of a function is symmetric with respect to the origin if for each point (a, b) on the graph, (−a, −b) is also a point on the graph.
a. Use a graphing calculator to graph the function y = xn when n = 1, 2, 3, 4, 5, and 6. In each case, identify the symmetry of the graph.
b. Predict what symmetry the graphs of y = x10 and y = x11 each have. Explain your reasoning and then confirm your predictions by graphing.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 47.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 47.2

Question 48.
HOW DO YOU SEE IT? The graph of a polynomial function is shown.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 14
a. Describe the degree and leading coefficient of f.
b. Describe the intervals where the function is increasing and decreasing.
c. What is the constant term of the polynomial function?
Answer:

Question 49.
REASONING A cubic polynomial function f has a leading coefficient of 2 and a constant term of −5. When f(1) = 0 and f(2) = 3, what is f(−5)? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 49

Question 50.
CRITICAL THINKING The weight y (in pounds) of a rainbow trout can be modeled by y = 0.000304x3, where x is the length (in inches) of the trout.
a. Write a function that relates the weight y and length x of a rainbow trout when y is measured in kilograms and x is measured in centimeters. Use the fact that 1 kilogram ≈ 2.20 pounds and 1 centimeter ≈ 0.394 inch.
b. Graph the original function and the function from part (a) in the same coordinate plane. What type of transformation can you apply to the graph of y = 0.000304x3 to produce the graph from part (a)?
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 15
Answer:

Maintaining Mathematical Proficiency

Simplify the expression. (Skills Review Handbook)

Question 51.
xy + x2 + 2xy + y2 − 3x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 51

Question 52.
2h3g + 3hg3 + 7h2g2 + 5h3g+ 2hg3
Answer:

Question 53.
−wk + 3kz − 2kw + 9zk − kw
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 53

Question 54.
a2(m − 7a3) − m(a2 − 10)
Answer:

Question 55.
3x(xy − 4) + 3(4xy + 3) − xy(x2y − 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.1 Question 55

Question 56.
cv(9 − 3c) + 2c(v − 4c) + 6c
Answer:

Lesson 4.2 Adding, Subtracting, and Multiplying Polynomials

Essential Question
How can you cube a binomial?

EXPLORATION 1
Cubing Binomials
Work with a partner.
Find each product. Show your steps.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 16

EXPLORATION 2
Generalizing Patterns for Cubing a Binomial
Work with a partner.

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 17
a. Use the results of Exploration 1 to describe a pattern for the coefficients of the terms when you expand the cube of a binomial. How is your pattern related to Pascal’s Triangle, shown at the right?
b. Use the results of Exploration 1 to describe a pattern for the exponents of the terms in the expansion of a cube of a binomial.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 18
c. Explain how you can use the patterns you described in parts (a) and (b) to find the product (2x − 3)3. Then find this product.

Communicate Your Answer

Question 3.
How can you cube a binomial?

Question 4.
Find each product.
a. (x + 2)3
b. (x − 2)3
c. (2x − 3)3
d. (x − 3)3
e. (−2x + 3)3
f. (3x − 5)3

4.2 Lesson

Monitoring Progress

Find the sum or difference.

Question 1.
(2x2 − 6x + 5) + (7x2 − x − 9)

Question 2.
(3t3 + 8t2 − t − 4) − (5t3 − t2 + 17)

Find the product.

Question 3.
(4x2 + x − 5)(2x + 1)

Question 4.
(y − 2)(5y2 + 3y − 1)

Question 5.
(m − 2)(m − 1)(m + 3)

Question 6.
(3t − 2)(3t + 2)

Question 7.
(5a + 2)2

Question 8.
(xy − 3)3

Question 9.
(a) Prove the polynomial identity for the cube of a binomial representing a difference: (a − b)3 = a3 − 3a2b + 3ab2 − b3.
(b) Use the cube of a binomial in part (a) to calculate 93.

Question 10.
Use Pascal’s Triangle to expand
(a) (z + 3)4 and
(b) (2t − 1)5

Adding, Subtracting, and Multiplying Polynomials 4.2 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING Describe three different methods to expand (x + 3)3.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 1

Question 2.
WRITING Is (a + b)(a − b) = a2 − b2 an identity? Explain your reasoning.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, find the sum.

Question 3.
(3x2 + 4x − 1) + (−2x2 − 3x + 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 3

Question 4.
(−5x2 + 4x − 2) + (−8x2 + 2x + 1)
Answer:

Question 5.
(12x5 − 3x4 + 2x − 5) + (8x4 − 3x3 + 4x + 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 5

Question 6.
(8x4 + 2x2 − 1) + (3x3 − 5x2 + 7x + 1)
Answer:

Question 7.
(7x6 + 2x5 − 3x2 + 9x) + (5x5 + 8x3 − 6x2 + 2x − 5)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 7

Question 8.
(9x4 − 3x3 + 4x2 + 5x + 7) + (11x4 − 4x2 − 11x − 9)
Answer:

In Exercises 9–14, find the difference.

Question 9.
(3x3 − 2x2 + 4x − 8) − (5x3 + 12x2 − 3x − 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 9

Question 10.
(7x4 − 9x3 − 4x2 + 5x + 6) − (2x4 + 3x3 − x2 + x − 4)
Answer:

Question 11.
(5x6 − 2x4 + 9x3 + 2x − 4) − (7x5 − 8x4 + 2x− 11)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 11

Question 12.
(4x5 − 7x3 − 9x2 + 18) − (14x5 − 8x4 + 11x2 + x)
Answer:

Question 13.
(8x5 + 6x3 − 2x2 + 10x) − (9x5 − x3 − 13x2 + 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 13

Question 14.
(11x4 − 9x2 + 3x + 11) − (2x4 + 6x3 + 2x − 9)
Answer:

Question 15.
MODELING WITH MATHEMATICS During a recent period of time, the numbers (in thousands) of males Mand females F that attend degree-granting institutions in the United States can be modeled by
M = 19.7t2 + 310.5t + 7539.6
F = 28t2 + 368t + 10127.8
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 19
where t is time in years. Write a polynomial to model the total number of people attending degree-granting institutions. Interpret its constant term.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 15

Question 16.
MODELING WITH MATHEMATICS A farmer plants a garden that contains corn and pumpkins. The total area (in square feet) of the garden is modeled by the expression 2x2 + 5x + 4. The area of the corn is modeled by the expression x2 − 3x + 2. Write an expression that models the area of the pumpkins.
Answer:

In Exercises 17–24, find the product.

Question 17.
7x3(5x2 + 3x + 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 17

Question 18.
−4x5(11x3 + 2x2 + 9x + 1)
Answer:

Question 19.
(5x2 − 4x + 6)(−2x + 3)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 19

Question 20.
(−x − 3)(2x2 + 5x + 8)
Answer:

Question 21.
(x2 − 2x − 4)(x2 − 3x − 5)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 21

Question 22.
(3x2 + x − 2)(−4x2 − 2x − 1)
Answer:

Question 23.
(3x3 − 9x + 7)(x2 − 2x + 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 23

Question 24.
(4x2 − 8x − 2)(x4 + 3x2 + 4x)
Answer:

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in performing the operation.

Question 25.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 20
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 25

Question 26.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 21
Answer:

In Exercises 27–32, find the product of the binomials.

Question 27.
(x − 3)(x + 2)(x + 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 27

Question 28.
(x − 5)(x + 2)(x − 6)
Answer:

Question 29.
(x − 2)(3x + 1)(4x − 3)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 29

Question 30.
(2x + 5)(x − 2)(3x + 4)
Answer:

Question 31.
(3x − 4)(5 − 2x)(4x + 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 31

Question 32.
(4 − 5x)(1 − 2x)(3x + 2)
Answer:

Question 33.
REASONING Prove the polynomial identity (a + b)(a − b) = a2 − b2. Then give an example of two whole numbers greater than 10 that can be multiplied using mental math and the given identity. Justify your answers.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 33

Question 34.
NUMBER SENSE You have been asked to order textbooks for your class. You need to order 29 textbooks that cost $31 each. Explain how you can use the polynomial identity (a + b)(a − b) = a2 − b2 and mental math to find the total cost of the textbooks.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 22
Answer:

In Exercises 35–42, find the product.

Question 35.
(x − 9)(x + 9)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 35

Question 36.
(m + 6)2
Answer:

Question 37.
(3c − 5)2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 37

Question 38.
(2y − 5)(2y + 5)
Answer:

Question 39.
(7h + 4)2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 39

Question 40.
(9g − 4)2
Answer:

Question 41.
(2k + 6)3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 41

Question 42.
(4n − 3)3
Answer:

In Exercises 43–48, use Pascal’s Triangle to expand the binomial.

Question 43.
(2t + 4)3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 43

Question 44.
(6m + 2)2
Answer:

Question 45.
(2q − 3)4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 45

Question 46.
(g + 2)5
Answer:

Question 47.
(yz + 1)5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 47

Question 48.
(np − 1)4
Answer:

Question 49.
COMPARING METHODS Find the product of the expression (a2 + 4b2)2(3a2 − b2)2 using two different methods. Which method do you prefer? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 49

Question 50.
THOUGHT PROVOKING Adjoin one or more polygons to the rectangle to form a single new polygon whose perimeter is double that of the rectangle. Find the perimeter of the new polygon.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 23
Answer:

MATHEMATICAL CONNECTIONS In Exercises 51 and 52, write an expression for the volume of the figure as a polynomial in standard form.

Question 51.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 24
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 51

Question 52.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 25
Answer:

Question 53.
MODELING WITH MATHEMATICS Two people make three deposits into their bank accounts earning the same simple interest rate r.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 26
Person A’s account is worth 2000(1 + r)3 + 3000(1 + r)2 + 1000(1 + r) on January 1, 2015.
a. Write a polynomial for the value of Person B’s account on January 1, 2015.
b. Write the total value of the two accounts as a polynomial in standard form. Then interpret the coefficients of the polynomial.
c. Suppose their interest rate is 0.05. What is the total value of the two accounts on January 1, 2015?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 53.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 53.2

Question 54.
PROBLEM SOLVING
The sphere is centered in the cube. Find an expression for the volume of the cube outside the sphere.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 27
Answer:

Question 55.
MAKING AN ARGUMENT Your friend claims the sum of two binomials is always a binomial and the product of two binomials is always a trinomial. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 55

Question 56.
HOW DO YOU SEE IT? You make a tin box by cutting x-inch-by-x-inch pieces of tin off the corners of a rectangle and folding up each side. The plan for your box is shown.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 28
a. What are the dimensions of the original piece of tin?
b. Write a function that represents the volume of the box. Without multiplying, determine its degree.
Answer:

USING TOOLS In Exercises 57–60, use a graphing calculator to make a conjecture about whether the two functions are equivalent. Explain your reasoning.

Question 57.
f(x) = (2x − 3)3; g(x) = 8x3 − 36x2 + 54x − 27
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 57

Question 58.
h(x) = (x + 2)5;
k(x) = x5 + 10x4 + 40x3 + 80x2 + 64x
Answer:

Question 59.
f(x) = (−x − 3)4;
g(x) = x4 + 12x3 + 54x2 + 108x + 80
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 59

Question 60.
f(x) = (−x + 5)3; g(x) = −x3 + 15x2 − 75x + 125
Answer:

Question 61.
REASONING Copy Pascal’s Triangle and add rows for n = 6, 7, 8, 9, and 10. Use the new rows to expand (x + 3)7 and (x − 5)9.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 61.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 61.2

Question 62.
ABSTRACT REASONING You are given the function f(x) = (x + a)(x + b)(x + c)(x + d). When f(x) is written in standard form, show that the coefficient of x3 is the sum of a, b, c, and d, and the constant term is the product of a, b, c, and d.
Answer:

Question 63.
DRAWING CONCLUSIONS Let g(x) = 12x4 + 8x + 9 and h(x) = 3x5 + 2x3 − 7x + 4.
a. What is the degree of the polynomial g(x) + h(x)?
b. What is the degree of the polynomial g(x) − h(x)?
c. What is the degree of the polynomial g(x) • h(x)?
d. In general, if g(x) and h(x) are polynomials such that g(x) has degree m and h(x) has degree n, and m > n, what are the degrees of g(x) + h(x), g(x) − h(x), and g(x) • h(x)?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 63.1

Question 64.
FINDING A PATTERN In this exercise, you will explore the sequence of square numbers. The first four square numbers are represented below.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 29
a. Find the differences between consecutive square numbers. Explain what you notice.
b. Show how the polynomial identity (n + 1)2 − n2 = 2n + 1 models the differences between square numbers.
c. Prove the polynomial identity in part (b).
Answer:

Question 65.
CRITICAL THINKING Recall that a Pythagorean triple is a set of positive integers a, b, and c such that a2 + b2 = c2. The numbers 3, 4, and 5 form a Pythagorean triple because 32 + 42 = 52. You can use the polynomial identity (x2 − y2)2 + (2xy)2 = (x2 + y2)2 to generate other Pythagorean triples.
a. Prove the polynomial identity is true by showing that the simplified expressions for the left and right sides are the same. b. Use the identity to generate the Pythagorean triple when x = 6 and y = 5.
c. Verify that your answer in part (b) satisfies a2 + b2 = c2.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 65.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 65.2

Maintaining Mathematical Proficiency

Perform the operation. Write the answer in standard form. (Section 3.2)

Question 66.
(3 − 2i) + (5 + 9i)
Answer:

Question 67.
(12 + 3i) − (7 − 8i)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 67

Question 68.
(7i)(−3i)
Answer:

Question 69.
(4 + i)(2 − i)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.2 Question 69

Lesson 4.3 Dividing Polynomials

Essential Question
How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial?

EXPLORATION 1
Dividing Polynomials
Work with a partner.
Match each division statement with the graph of the related cubic polynomial f(x). Explain your reasoning. Use a graphing calculator to verify your answers.
a. \(\frac{f(x)}{x}\) = (x − 1)(x + 2)
b. \(\frac{f(x)}{x-1}\) = (x − 1)(x + 2)
c. \(\frac{f(x)}{x+1}\) = (x − 1)(x + 2)
d. \(\frac{f(x)}{x-2}\) = (x− 1)(x+ 2)
e. \(\frac{f(x)}{x+2}\) = (x− 1)(x+ 2)
f. \(\frac{f(x)}{x-3}\) = (x− 1)(x+ 2)
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 29.1

EXPLORATION 2
Dividing Polynomials

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 30
Work with a partner. Use the results of Exploration 1 to find each quotient. Write your answers in standard form. Check your answers by multiplying.
a. (x3 + x2 − 2x) ÷ x
b. (x3 − 3x + 2) ÷ (x − 1)
c. (x3 + 2x2 − x − 2) ÷ (x + 1)
d. (x3 − x2 − 4x + 4) ÷ (x − 2)
e. (x3 + 3x2 − 4) ÷ (x + 2)
f. (x3 − 2x2 − 5x + 6) ÷ (x − 3)

Communicate Your Answer

Question 3.
How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial?

4.3 Lesson

Monitoring Progress

Divide using polynomial long division.

Question 1.
(x3 − x2 − 2x + 8) ÷ (x − 1)

Question 2.
(x4 + 2x2 − x + 5) ÷ (x2 − x + 1)

Divide using synthetic division.

Question 3.
(x3 − 3x2 − 7x + 6) ÷ (x − 2)

Question 4.
(2x3 − x − 7) ÷ (x + 3)

Use synthetic division to evaluate the function for the indicated value of x.

Question 5.
f(x) = 4x2 − 10x − 21; x = 5

Question 6.
f(x) = 5x4 + 2x3 − 20x − 6; x = 2

Dividing Polynomials 4.3 Exercises

Monitoring Progress and Modeling with Mathematics

Question 1.
WRITING Explain the Remainder Theorem in your own words. Use an example in your explanation.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 1

Question 2.
VOCABULARY What form must the divisor have to make synthetic division an appropriate method for dividing a polynomial? Provide examples to support your claim.
Answer:

Question 3.
VOCABULARY Write the polynomial divisor, dividend, and quotient functions represented by the synthetic division shown at the right.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 31
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 3

Question 4.
WRITING Explain what the colored numbers represent in the synthetic division in Exercise 3.
Answer:

Vocabulary and Core Concept Check

In Exercises 5–10, divide using polynomial long division.

Question 5.
(x2 + x − 17 ) ÷ (x − 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 5

Question 6.
(3x2 − 14x − 5) ÷ (x − 5)
Answer:

Question 7.
(x3 + x2 + x + 2 ) ÷ (x2 − 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 7

Question 8.
(7x3 + x2 + x ) ÷ (x2 + 1)
Answer:

Question 9.
(5x4 − 2x3 − 7x2 − 39) ÷ (x2 + 2x − 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 9

Question 10.
(4x4 + 5x − 4) ÷ (x2 − 3x − 2)
Answer:

In Exercises 11–18, divide using synthetic division.

Question 11.
(x2 + 8x + 1) ÷ (x − 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 11

Question 12.
(4x2 − 13x − 5) ÷ (x − 2)
Answer:

Question 13.
(2x2 − x + 7) ÷ (x + 5)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 13

Question 14.
(x3 − 4x + 6) ÷ (x + 3)
Answer:

Question 15.
(x2 + 9) ÷ (x − 3)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 15

Question 16.
(3x3 − 5x2 − 2) ÷ (x − 1)
Answer:

Question 17.
(x4 − 5x3 − 8x2 + 13x − 12) ÷ (x − 6)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 17

Question 18.
(x4 + 4x3 + 16x − 35) ÷ (x + 5 )
Answer:

ANALYZING RELATIONSHIPS In Exercises 19–22, match the equivalent expressions. Justify your answers.

Question 19.
(x2 + x − 3) ÷ (x − 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 19

Question 20.
(x2 − x − 3) ÷ (x − 2)
Answer:

Question 21.
(x2 − x + 3) ÷ (x − 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 21

Question 22.
(x2 + x + 3) ÷ (x − 2)
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 32
Answer:

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in using synthetic division to divide x3 – 5x + 3 by x – 2.

Question 23.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 33
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 23

Question 24.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 34
Answer:

In Exercises 25–32, use synthetic division to evaluate the function for the indicated value of x.

Question 25.
f(x) = −x2 − 8x + 30; x = −1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 25

Question 26.
f(x) = 3x2 + 2x − 20; x = 3
Answer:

Question 27.
f(x) = x3 − 2x2 + 4x + 3; x = 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 27

Question 28.
f(x) = x3 + x2 − 3x + 9; x = −4
Answer:

Question 29.
f(x) = x3 − 6x + 1; x = 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 29

Question 30.
f(x) = x3 − 9x − 7; x = 10
Answer:

Question 31.
f(x) = x4 + 6x2 − 7x + 1; x = 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 31

Question 32.
f(x) = −x4 − x3 − 2; x = 5
Answer:

Question 33.
MAKING AN ARGUMENT You use synthetic division to divide f(x) by (x − a) and find that the remainder equals 15. Your friend concludes that f(15) = a. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 33

Question 34.
THOUGHT PROVOKING A polygon has an area represented by A = 4x2 + 8x + 4. The figure has at least one dimension equal to 2x + 2. Draw the figure and label its dimensions.
Answer:

Question 35.
USING TOOLS The total attendance A (in thousands) at NCAA women’s basketball games and the number T of NCAA women’s basketball teams over a period of time can be modeled by
A = −1.95x3 + 70.1x2 − 188x + 2150
T = 14.8x + 725
where x is in years and 0 < x < 18. Write a function for the average attendance per team over this period of time.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 35
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 35

Question 36.
COMPARING METHODS The profit P (in millions of dollars) for a DVD manufacturer can be modeled by P = −6x3 + 72x, where x is the number (in millions) of DVDs produced. Use synthetic division to show that the company yields a profit of $96 million when 2 million DVDs are produced. Is there an easier method? Explain.
Answer:

Question 37.
CRITICAL THINKING What is the value of k such that (x3 − x2 + kx − 30) ÷ (x − 5) has a remainder of zero?
A. −14
B. −2
C. 26
D. 32
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 37

Question 38.
HOW DO YOU SEE IT? The graph represents the polynomial function f(x) = x3 + 3x2 − x − 3.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 36
a. The expression f(x) ÷ (x − k) has a remainder of −15. What is the value of k?
b. Use the graph to compare the remainders of (x3 + 3x2 − x − 3) ÷ (x + 3) and (x3 + 3x2 − x − 3) ÷ (x + 1).
Answer:

Question 39.
MATHEMATICAL CONNECTIONS The volume V of the rectangular prism is given by V = 2x3 + 17x2 + 46x + 40. Find an expression for the missing dimension.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 37
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 39

Question 40.
USING STRUCTURE You divide two polynomials and obtain the result 5x2 − 13x + 47 − \(\frac{102}{x+2}\). What is the dividend? How did you find it?
Answer:

Maintaining Mathematical Proficiency

Find the zero(s) of the function. (Sections 3.1and 3.2)

Question 41.
f(x) = x2 − 6x + 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 41

Question 42.
g(x) = 3(x + 6)(x − 2)
Answer:

Question 43.
g(x) = x2 + 14x + 49
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.3 Question 43

Question 44.
h(x) = 4x2 + 36

Lesson 4.4 Factoring Polynomials

Essential Question
How can you factor a polynomial?

EXPLORATION 1
Factoring Polynomials
Work with a partner.
Match each polynomial equation with the graph of its related polynomial function. Use the x-intercepts of the graph to write each polynomial in factored form. Explain your reasoning.
a. x2 + 5x + 4 = 0
b. x3 − 2x2 − x + 2 = 0
c. x3 + x2 − 2x = 0
d. x3 − x = 0
e. x4 − 5x2 + 4 = 0
f. x4 − 2x3 − x2 + 2x = 0
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 38

EXPLORATION 2
Factoring Polynomials

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 39
Work with a partner. Use the x-intercepts of the graph of the polynomial function to write each polynomial in factored form. Explain your reasoning. Check your answers by multiplying.
a. f(x) = x2 − x − 2
b. f(x) = x3 − x2 − 2x
c. f(x) = x3 − 2x2 − 3x
d. f(x) = x3 − 3x2 − x + 3
e. f(x) = x4 + 2x3 − x2 − 2x
f. f(x) = x4 − 10x2 + 9

Communicate Your Answer

Question 3.
How can you factor a polynomial?

Question 4.
What information can you obtain about the graph of a polynomial function written in factored form?

4.4 Lesson

Monitoring Progress

Question 1.
x3 − 7x2 + 10x

Question 2.
3n7 − 75n5

Question 3.
8m5 − 16m4 + 8m3

Factor the polynomial completely

Question 4.
a3 + 27

Question 5.
6z5 − 750z2

Question 6.
x3 + 4x2 − x − 4

Question 7.
3y3 + y2 + 9y + 3

Question 8.
−16n4 + 625

Question 9.
5w6 − 25w4 + 30w2

Question 10.
Determine whether x− 4 is a factor of f(x) = 2x2 + 5x − 12.

Question 11.
Show that x − 6 is a factor of f(x) = x3 − 5x2 − 6x. Then factor f(x) completely.

Question 12.
In Example 7, does your answer change when you first determine whether 2 is a zero and then whether −1 is a zero? Justify your answer.

Factoring Polynomials 4.4 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE The expression 9x4 − 49 is in _________ form because it can be written as u2 − 49 where u = _____.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 1

Question 2.
VOCABULARY Explain when you should try factoring a polynomial by grouping.
Answer:

Question 3.
WRITING How do you know when a polynomial is factored completely?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 3

Question 4.
WRITING Explain the Factor Theorem and why it is useful.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, factor the polynomial completely.

Question 5.
x3 − 2x2 − 24x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 5

Question 6.
4k5 − 100k3
Answer:

Question 7.
3p5 − 192p3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 7

Question 8.
2m6 − 24m5 + 64m4
Answer:

Question 9.
2q4 + 9q3 − 18q2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 9

Question 10.
3r6 − 11r5 − 20r4
Answer:

Question 11.
10w10 − 19w9 + 6w8
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 11

Question 12.
18v9 + 33v8 + 14v7
Answer:

In Exercises 13–20, factor the polynomial completely.

Question 13.
x3 + 64
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 13

Question 14.
y3 + 512
Answer:

Question 15.
g3 − 343
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 15

Question 16.
c3 − 27
Answer:

Question 17.
3h9 − 192h6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 17

Question 18.
9n6 − 6561n3
Answer:

Question 19.
16t7 + 250t4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 19

Question 20.
135z11 − 1080z8
Answer:

ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in factoring the polynomial.

Question 21.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 40
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 21

Question 22.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 41
Answer:

In Exercises 23–30, factor the polynomial completely.

Question 23.
y3 − 5y2 + 6y − 30
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 23

Question 24.
m3 − m2 + 7m − 72
Answer:

Question 25.
3a3 + 18a2 + 8a + 48
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 25

Question 26.
2k3 − 20k2 + 5k − 50
Answer:

Question 27.
x3 − 8x2 − 4x + 32
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 27

Question 28.
z3 − 5z2 − 9z + 45
Answer:

Question 29.
4q3 − 16q7 − 9q + 36
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 29

Question 30.
16n3 + 32n7 − n − 2
Answer:

In Exercises 31–38, factor the polynomial completely.

Question 31.
49k4 − 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 31

Question 32.
4m4 − 25
Answer:

Question 33.
c4 + 9c2 + 20
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 33

Question 34.
y4 − 3y2 − 28
Answer:

Question 35.
16z4 − 81
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 35

Question 36.
81a4 − 256
Answer:

Question 37.
3r8 + 3r5 − 60r2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 37

Question 38.
4n12 − 32n7 + 48n2
Answer:

In Exercises 39–44, determine whether the binomial is a factor of the polynomial.

Question 39.
f(x) = 2x3 + 5x2 − 37x − 60; x − 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 39

Question 40.
g(x) = 3x3 − 28x2 + 29x + 140; x + 7
Answer:

Question 41.
h(x) = 6x5 − 15x4 − 9x3; x + 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 41

Question 42.
g(x) = 8x5 − 58x4 + 60x3 + 140; x − 6
Answer:

Question 43.
h(x) = 6x4 − 6x3 − 84x2 + 144x; x + 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 43

Question 44.
t(x) = 48x4 + 36x3 − 138x2 − 36x; x + 2
Answer:

In Exercises 45–50, show that the binomial is a factor of the polynomial. Then factor the polynomial completely.

Question 45.
g(x) = x3 − x2 − 20x; x + 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 45

Question 46.
t(x) = x3 − 5x2 − 9x + 45; x − 5
Answer:

Question 47.
f(x) = x4 − 6x3 − 8x + 48; x − 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 47

Question 48.
s(x) = x4 + 4x3 − 64x − 256; x + 4
Answer:

Question 49.
r(x) = x3 − 37x + 84; x + 7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 49

Question 50.
h(x) = x3 − x2 − 24x − 36; x + 2
Answer:

ANALYZING RELATIONSHIPS In Exercises 51–54, match the function with the correct graph. Explain your reasoning.

Question 51.
f(x) = (x − 2)(x − 3)(x + 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 51

Question 52.
g(x) = x(x + 2)(x + 1)(x − 2)
Answer:

Question 53.
h(x) = (x + 2)(x + 3)(x − 1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 53

Question 54.
k(x) = x(x − 2)(x − 1)(x + 2)
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 42
Answer:

Question 55.
MODELING WITH MATHEMATICS The volume (in cubic inches) of a shipping box is modeled by V = 2x3 − 19x2 + 39x, where x is the length (in inches). Determine the values of x for which the model makes sense. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 43
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 55

Question 56.
MODELING WITH MATHEMATICS The volume (in cubic inches) of a rectangular birdcage can be modeled by V = 3x3 − 17x2 + 29x − 15, where x is the length (in inches). Determine the values of x for which the model makes sense. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 44
Answer:

USING STRUCTURE In Exercises 57–64, use the method of your choice to factor the polynomial completely. Explain your reasoning.

Question 57.
a6 + a5 − 30a4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 57

Question 58.
8m3 − 343
Answer:

Question 59.
z3 − 7z2 − 9z + 63
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 59

Question 60.
2p8 − 12p5 + 16p2
Answer:

Question 61.
64r3 + 729
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 61

Question 62.
5x5 − 10x4 − 40x3
Answer:

Question 63.
16n4 − 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 63

Question 64.
9k3 − 24k2 + 3k − 8
Answer:

Question 65.
REASONING Determine whether each polynomial is factored completely. If not, factor completely.
a. 7z4(2z2 − z − 6)
b. (2 − n)(n2 + 6n)(3n − 11)
c. 3(4y − 5)(9y2 − 6y − 4)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 65

Question 66.
PROBLEM SOLVING The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled by P = −x3 + 4x2 + x, where x is the number (in millions) of T-shirts produced. Currently the company produces 4 million T-shirts and makes a profit of $4 million. What lesser number of T-shirts could the company produce and still make the same profit?
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 44.1
Answer:

Question 67.
PROBLEM SOLVING The profit P (in millions of dollars) for a shoe manufacturer can be modeled by P = −21x3 + 46x, where x is the number (in millions) of shoes produced. The company now produces 1 million shoes and makes a profit of $25 million, but it would like to cut back production. What lesser number of shoes could the company produce and still make the same profit?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 67

Question 68.
THOUGHT PROVOKING Find a value of k such that \(\frac{f(x)}{x-k}\) has a remainder of 0. Justify your answer.
Answer:

Question 69.
COMPARING METHODS You are taking a test where calculators are not permitted. One question asks you to evaluate g(7) for the function g(x) = x3 − 7x2 − 4x + 28. You use the Factor Theorem and synthetic division and your friend uses direct substitution. Whose method do you prefer? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 69

Question 70.
MAKING AN ARGUMENT You divide f(x) by (x−a) and find that the remainder does not equal 0. Your friend concludes that f(x) cannot be factored. Is your friend correct? Explain your reasoning.
Answer:

Question 71.
CRITICAL THINKING What is the value of k such that x− 7 is a factor of h(x) = 2x3 − 13x2 − kx + 105? Justify your answer.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 71

Question 72.
HOW DO YOU SEE IT? Use the graph to write an equation of the cubic function in factored form. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 45
Answer:

Question 73.
ABSTRACT REASONING Factor each polynomial completely.
a. 7ac2 + bc2 − 7ad2 − bd2
b. x2n − 2xn + 1
c. a5b2 − a2b4 + 2a4b − 2ab3 + a3 − b2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 73

Question 74.
REASONING The graph of the function f(x) = x4 + 3x3 + 2x2 + x + 3 is shown. Can you use the Factor Theorem to factor f(x)? Explain.
Answer:

Question 75.
MATHEMATICAL CONNECTIONS The standard equation of a circle with radius r and center (h, k) is (x − h)2 + (y − k)2 = r2. Rewrite each equation of a circle in standard form. Identify the center and radius of the circle. Then graph the circle.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 46
a. x2 + 6x + 9 + y2 = 25
b. x2 − 4x + 4 + y2 = 9
c. x2 − 8x + 16 + y2 + 2y + 1 = 36
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 75.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 75.2
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 75.3

Question 76.
CRITICAL THINKING Use the diagram to complete parts (a)–(c).
a. Explain why a3 − b3 is equal to the sum of the volumes of the solids I, II, and III.
b. Write an algebraic expression for the volume of each of the three solids. Leave your expressions in factored form.
c. Use the results from part (a) and part (b) to derive the factoring pattern a3 − b3.
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 47
Answer:

Maintaining Mathematical Proficiency

Solve the quadratic equation by factoring.(Section 3.1)

Question 77.
x2 − x − 30 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 77

Question 78.
2x2 − 10x − 72 = 0
Answer:

Question 79.
3x2 − 11x + 10 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 79

Question 80.
9x2 − 28x + 3 = 0
Answer:

Solve the quadratic equation by completing the square.(Section 3.3)

Question 81.
x2 − 12x + 36 = 144
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 81

Question 82.
x2 − 8x − 11 = 0
Answer:

Question 83.
3x2 + 30x + 63 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.4 Question 83

Question 84.
4x2 + 36x − 4 = 0
Answer:

Polynomial Functions Study Skills : Keeping Your Mind Focused

4.1–4.4 What Did You Learn?

Core Vocabulary
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 48

Core Concepts
Section 4.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 49

Section 4.2
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 50

Section 4.3
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 51

Section 4.4
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 52

Mathematical Practices

Question 1.
Describe the entry points you used to analyze the function in Exercise 43 on page 164.

Question 2.
Describe how you maintained oversight in the process of factoring the polynomial in Exercise 49 on page 185.

Study Skills
Keeping Your Mind Focused

  • When you sit down at your desk, review your notes from the last class.
  • Repeat in your mind what you are writing in your notes.
  • When a mathematical concept is particularly difficult, ask your teacher for another example.

Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 52.1

Polynomial Functions 4.1 – 4.4 Quiz

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.(Section 4.1)

Question 1.
f(x) = 5 + 2x2 − 3x4 − 2x − x3

Question 2.
g(x) = \(\frac{1}{4}\)x3 + 2x − 3x2 + 1

Question 3.
h(x) = 3 − 6x3 + 4x − 2 + 6x

Question 4.
Describe the x-values for which
(a) f is increasing or decreasing,
(b) f(x) > 0, and
(c) f(x) < 0. (Section 4.1)
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 53

Question 5.
Write an expression for the area and perimeter for the figure shown.(Section 4.2)
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 54

Perform the indicated operation.(Section 4.2)

Question 6.
(7x2 − 4) − (3x2 − 5x + 1)

Question 7.
(x2 − 3x + 2)(3x − 1)

Question 8.
(x − 1)(x + 3)(x − 4)

Question 9.
Use Pascal’s Triangle to expand (x + 2)5. (Section 4.2)

Question 10.
Divide 4x4 − 2x3 + x2 − 5x + 8 by x2 − 2x − 1. (Section 4.3)

Factor the polynomial completely.(Section 4.4)

Question 11.
a3 − 2a2 − 8a

Question 12.
8m3 + 27

Question 13.
z3 + z2 − 4z − 4

Question 14.
49b4 − 64

Question 15.
Show that x + 5 is a factor of f(x) = x3 − 2x2 − 23x + 60. Then factor f(x) completely. (Section 4.4)

Question 16.
The estimated price P (in cents) of stamps in the United States can be modeled by the polynomial function P(t)= 0.007t3 − 0.16t2 + 1t + 17, where t represents the number of years since 1990. (Section 4.1)
a. Use a graphing calculator to graph the function for the interval 0 ≤ t ≤ 20. Describe the behavior of the graph on this interval.
b. What was the average rate of change in the price of stamps from 1990 to 2010?

Question 17.
The volume V (in cubic feet) of a rectangular wooden crate is modeled by the function V(x) = 2x3 − 11x2 + 12x, where x is the width (in feet) of the crate. Determine the values of x for which the model makes sense. Explain your reasoning. (Section 4.4)
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 55

Lesson 4.5 Solving Polynomial Equations

Essential Question
How can you determine whether a polynomial equation has a repeated solution?

EXPLORATION 1
Cubic Equations and Repeated Solutions
Work with a partner.
Some cubic equations have three distinct solutions. Others have repeated solutions. Match each cubic polynomial equation with the graph of its related polynomial function. Then solve each equation. For those equations that have repeated solutions, describe the behavior of the related function near the repeated zero using the graph or a table of values.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 56
a. x3 − 6x2 + 12x − 8 = 0
b. x3 + 3x2 + 3x + 1 = 0
c. x3− 3x + 2 = 0
d. x3 + x2 − 2x = 0
e. x3 − 3x − 2 = 0
f. x3 − 3x2 + 2x = 0
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 57

EXPLORATION 2
Quartic Equations and Repeated Solutions
Work with a partner.
Determine whether each quartic equation has repeated solutions using the graph of the related quartic function or a table of values. Explain your reasoning. Then solve each equation.
a. x4 − 4x3 + 5x2 − 2x = 0
b. x4 − 2x3 − x2 + 2x = 0
c. x4 − 4x3 + 4x2 = 0
d. x4 + 3x3 = 0

Communicate Your Answer

Question 3.
How can you determine whether a polynomial equation has a repeated solution?

Question 4.
Write a cubic or a quartic polynomial equation that is different from the equations in Explorations 1 and 2 and has a repeated solution.

4.5 Lesson

Monitoring Progress

Solve the equation.

Question 1.
4x4 − 40x2 + 36 = 0

Question 2.
2x5 + 24x = 14x3

Find the zeros of the function. Then sketch a graph of the function.

Question 3.
f(x) = 3x4 − 6x2 + 3

Question 4.
f(x) = x3 + x2 − 6x

Question 5.
Find all real solutions of x3 − 5x2 − 2x + 24 = 0.

Question 6.
Find all real zeros of f(x) = 3x4 − 2x3 − 37x2 + 24x + 12.

Question 7.
Write a polynomial function f of least degree that has rational coefficients, aleading coefficient of 1, and the zeros 4 and 1 − \(\sqrt{5}\).

Solving Polynomial Equations 4.5 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE If a polynomial function f has integer coefficients, then every rational solution of f(x) = 0 has the form \(\frac{p}{q}\), where p is a factor of the _____________ and q is a factor of the _____________.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 1

Question 2.
DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 58
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, solve the equation.

Question 3.
z3 − z2 − 12z = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 3

Question 4.
a3 − 4a2 + 4a = 0
Answer:

Question 5.
2x4 − 4x3 = −2x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 5

Question 6.
v3 − 2v2 − 16v = − 32
Answer:

Question 7.
5w3 = 50w
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 7

Question 8.
9m5 = 27m3
Answer:

Question 9.
2c4 − 6c3 = 12c2 − 36c
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 9

Question 10.
p4 + 40 = 14p2
Answer:

Question 11.
12n2 + 48n = −n3 − 64
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 11

Question 12.
y3 − 27 = 9y2 − 27y
Answer:

In Exercises 13–20, find the zeros of the function. Then sketch a graph of the function.

Question 13.
h(x) = x4 + x3 − 6x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 13

Question 14.
f(x) = x4 − 18x2 + 81
Answer:

Question 15.
p(x) = x6 − 11x5 + 30x4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 15

Question 16.
g(x) = −2x5 + 2x4 + 40x3
Answer:

Question 17.
g(x) = −4x4 + 8x3 + 60x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 17

Question 18.
h(x) = −x3 − 2x2 + 15x
Answer:

Question 19.
h(x) = −x3 − x2 + 9x + 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 19

Question 20.
p(x) = x3 − 5x2 − 4x + 20
Answer:

Question 21.
USING EQUATIONS According to the Rational Root Theorem, which is not a possible solution of the equation 2x4 − 5x3 + 10x2 − 9 = 0?
A. −9
B. −\([\frac{1}{2}/latex]
C. [latex][\frac{5}{2}/latex]
D. 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 21

Question 22.
USING EQUATIONS According to the Rational Root Theorem, which is not a possible zero of the function f(x) = 40x5 − 42x4 − 107x3 + 107x2 + 33x − 36?
A. −[latex][\frac{2}{3}/latex]
B. − [latex][\frac{3}{8}/latex]
C. [latex][\frac{3}{4}/latex]
D. [latex][\frac{4}{5}/latex]
Answer:

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in listing the possible rational zeros of the function.

Question 23.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 59
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 23

Question 24.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 60
Answer:

In Exercises 25–32, find all the real solutions of the equation.

Question 25.
x3 + x2 − 17x + 15 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 25

Question 26.
x3 − 2x2 − 5x + 6 = 0
Answer:

Question 27.
x3 − 10x2 + 19x + 30 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 27

Question 28.
x3 + 4x2 − 11x − 30 = 0
Answer:

Question 29.
x3 − 6x2 − 7x + 60 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 29

Question 30.
x3 − 16x2 + 55x + 72 = 0
Answer:

Question 31.
2x3 − 3x2 − 50x − 24 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 31

Question 32.
3x3 + x2 − 38x + 24 = 0
Answer:

In Exercises 33–38, find all the real zeros of the function.

Question 33.
f(x) = x3 − 2x2 − 23x + 60
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 33

Question 34.
g(x) = x3 − 28x − 48
Answer:

Question 35.
h(x) = x3 + 10x2 + 31x + 30
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 35

Question 36.
f(x) = x3 − 14x2 + 55x − 42
Answer:

Question 37.
p(x) = 2x3 −x2 − 27x + 36
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 37

Question 38.
g(x) = 3x3 − 25x2 + 58x − 40
Answer:

USING TOOLS In Exercises 39 and 40, use the graph to shorten the list of possible rational zeros of the function. Then find all real zeros of the function.

Question 39.
f(x) = 4x3 − 20x + 16
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 61
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 39

Question 40.
f(x) = 4x3 − 49x − 60
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 62
Answer:

In Exercises 41–46, write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros.

Question 41.
−2, 3, 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 41

Question 42.
−4, −2, 5
Answer:

Question 43.
−2, 1 + [latex]\sqrt{7}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 43

Question 44.
4, 6 − \(\sqrt{7}\)
Answer:

Question 45.
−6, 0, 3 −\(\sqrt{5}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 45

Question 46.
0, 5, −5 + \(\sqrt{8}\)
Answer:

Question 47.
COMPARING METHODS Solve the equation x3 − 4x2 − 9x + 36 = 0 using two different methods. Which method do you prefer? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 47.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 47.2

Question 48.
REASONING Is it possible for a cubic function to have more than three real zeros? Explain.
Answer:

Question 49.
PROBLEM SOLVING At a factory, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism with a height 3 centimeters greater than the length of each side of its square base. Each mold holds 112 cubic centimeters of glass. What are the dimensions of the mold?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 49

Question 50.
MATHEMATICAL CONNECTIONS The volume of the cube shown is 8 cubic centimeters.
a. Write a polynomial equation that you can use to find the value of x.
b. Identify the possible rational solutions of the equation in part (a).
c. Use synthetic division to find a rational solution of the equation. Show that no other real solutions exist.
d. What are the dimensions of the cube?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 63
Answer:

Question 51.
PROBLEM SOLVING Archaeologists discovered a huge hydraulic concrete block at the ruins of Caesarea with a volume of 945 cubic meters. The block is x meters high by 12x − 15 meters long by 12x − 21 meters wide. What are the dimensions of the block?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 64
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 51

Question 52.
MAKING AN ARGUMENT Your friend claims that when a polynomial function has a leading coefficient of 1 and the coefficients are all integers, every possible rational zero is an integer. Is your friend correct? Explain your reasoning.
Answer:

Question 53.
MODELING WITH MATHEMATICS During a 10-year period, the amount (in millions of dollars) of athletic equipment Esold domestically can be modeled by E(t) = −20t3 + 252t2 − 280t + 21,614, where t is in years.
a. Write a polynomial equation to find the year when about $24,014,000,000 of athletic equipment is sold.
b. List the possible whole-number solutions of the equation in part (a). Consider the domain when making your list of possible solutions.
c. Use synthetic division to find when $24,014,000,000 of athletic equipment is sold.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 53

Question 54.
THOUGHT PROVOKING Write a third or fourth degree polynomial function that has zeros at ± \(\frac{3}{4}\). Justify your answer.
Answer:

Question 55.
MODELING WITH MATHEMATICS You are designing a marble basin that will hold a fountain for a city park. The sides and bottom of the basin should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 65
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 55

Question 56.
HOW DO YOU SEE IT? Use the information in the graph to answer the questions.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 66
a. What are the real zeros of the function f ?
b. Write an equation of the quartic function in factored form.
Answer:

Question 57.
REASONING Determine the value of k for each equation so that the given x-value is a solution.
a. x3 − 6x2 − 7x + k = 0; x = 4
b. 2x3 + 7x2 − kx − 18 = 0; x = −6
c. kx3 − 35x2 + 19x + 30 = 0; x = 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 57

Question 58.
WRITING EQUATIONS Write a polynomial function gof least degree that has rational coefficients, a leading coefficient of 1, and the zeros −2 + \(\sqrt{7}\) and 3 + \(\sqrt{2}\).
Answer:

In Exercises 59–62, solve f(x) = g(x) by graphing and algebraic methods.

Question 59.
f(x) = x3 + x2 − x − 1; g(x) = −x + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 59

Question 60.
f(x) = x4 − 5x3 + 2x2 + 8x; g(x) = −x2 + 6x − 8
Answer:

Question 61.
f(x) = x3 − 4x2 + 4x; g(x) = −2x + 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 61

Question 62.
f(x) = x4 + 2x3 − 11x2 − 12x + 36;
g(x) = −x2 − 6x − 9
Answer:

Question 63.
MODELING WITH MATHEMATICS You are building a pair of ramps for a loading platform. The left ramp is twice as long as the right ramp. If 150 cubic feet of concrete are used to build the ramps, what are the dimensions of each ramp?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 67
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 63

Question 64.
MODELING WITH MATHEMATICS Some ice sculptures are made by filling a mold and then freezing it. You are making an ice mold for a school dance. It is to be shaped like a pyramid with a height 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 68
Answer:

Question 65.
ABSTRACT REASONING Let an be the leading coefficient of a polynomial function f and a0 be the constant term. If an has r factors and a0 has s factors, what is the greatest number of possible rational zeros of f that can be generated by the Rational Zero Theorem? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 65

Maintaining Mathematical Proficiency

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.(Section 4.1)

Question 66.
h(x) = −3x2 + 2x − 9 + \(\sqrt{4}\)x3
Answer:

Question 67.
g(x) = 2x3 − 7x2 − 3x − 1 + x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 67

Question 68.
f(x) = \(\frac{1}{3}\)x2 + 2x3 − 4x4 − \(\sqrt{3}\)
Answer:

Question 69.
p(x) = 2x − 5x3 + 9x2 + \(\sqrt[4]{x}\) + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 69

Find the zeros of the function.(Section 3.2)

Question 70.
f(x) = 7x2 + 42
Answer:

Question 71.
g(x) = 9x2 + 81
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 71

Question 72.
h(x) = 5x2 + 40
Answer:

Question 73.
f(x) = 8x2 − 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.5 Question 73

Lesson 4.6 The Fundamental Theorem of Algebra

Essential Question

How can you determine whether a polynomial equation has imaginary solutions?

EXPLORATION 1
Cubic Equations and Imaginary Solutions
Work with a partner.
Match each cubic polynomial equation with the graph of its related polynomial function. Then find all solutions. Make a conjecture about how you can use a graph or table of values to determine the number and types of solutions of a cubic polynomial equation.
a. x3 − 3x2 + x + 5 = 0
b. x3 − 2x2 − x + 2 = 0
c. x3 − x2 − 4x + 4 = 0
d. x3 + 5x2 + 8x + 6 = 0
e. x3 − 3x2 + x − 3 = 0
f. x3 − 3x2 + 2x = 0
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 69

EXPLORATION 2
Quartic Equations and Imaginary Solutions
Work with a partner.
Use the graph of the related quartic function, or a table of values, to determine whether each quartic equation has imaginary solutions. Explain your reasoning. Then find all solutions.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 70
a. x4 − 2x3 − x2 + 2x = 0
b. x4 − 1 = 0
c. x4 + x3 − x − 1 = 0
d. x4 − 3x3 + x2 + 3x − 2 = 0

Communicate Your Answer

Question 3.
How can you determine whether a polynomial equation has imaginary solutions?

Question 4.
Is it possible for a cubic equation to have three imaginary solutions? Explain your reasoning.

4.6 Lesson

Monitoring Progress

Question 1.
How many solutions does the equation x4 + 7x2 − 144 = 0 have?

Question 2.
How many zeros does the function f(x) = x3 − 5x2 − 8x + 48 have? Find all zeros of the polynomial function.

Question 3.
f(x) = x3 + 7x2 + 16x + 12

Question 4.
f(x) = x5 − 3x4 + 5x3 − x2 − 6x + 4

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

Question 5.
−1, 4i

Question 6.
3, 1 + i\(\sqrt{5}\)

Question 7.
\(\sqrt{2}\), 1 − 3i

Question 8.
2, 2i, 4 − \(\sqrt{6}\)

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.

Question 9.
f(x) = x3 + 9x − 25

Question 10.
f(x) = 3x4 − 7x3 + x2 − 13x + 8

Question 11.
WHAT IF? In Example 5, what is the tachometer reading when the boat travels 20 miles per hour?

The Fundamental Theorem of Algebra 4.6 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE The expressions 5 + i and 5 − i are _____________.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 1

Question 2.
WRITING How many solutions does the polynomial equation (x + 8)3(x − 1) = 0 have? Explain.
Answer:

Monitoring Progress and Modeling with Mathematics

Question 3.
x4 + 2x3 − 4x2 + x = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 3

Question 4.
5y3 − 3y2 + 8y = 0
Answer:

Question 5.
9t6 − 14t3 + 4t − 1 = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 5

Question 6.
f(z) = −7z4 + z2 − 25
Answer:

Question 7.
g(s) = 4s5 − s3 + 2s7 − 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 7

Question 8.
h(x) = 5x4 + 7x8 − x12
Answer:

In Exercises 9–16, find all zeros of the polynomial function.

Question 9.
f(x) = x4 − 6x3 + 7x2 + 6x − 8
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 9

Question 10.
f(x) = x4 + 5x3 − 7x2 − 29x + 30
Answer:

Question 11.
g(x) = x4 − 9x2 − 4x + 12
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 11

Question 12.
h(x) = x3 + 5x2 − 4x − 20
Answer:

Question 13.
g(x) = x4 + 4x3 + 7x2 + 16x + 12
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 13

Question 14.
h(x) = x4 − x3 + 7x2 − 9x − 18
Answer:

Question 15.
g(x) = x5 + 3x4 − 4x3 − 2x2 − 12x − 16
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 15

Question 16.
f(x) = x5 − 20x3 + 20x2 − 21x + 20
Answer:

ANALYZING RELATIONSHIPS In Exercises 17–20, determine the number of imaginary zeros for the function with the given degree and graph. Explain your reasoning.

Question 17.
Degree: 4
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 70.1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 17

Question 18.
Degree: 5
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 71
Answer:

Question 19.
Degree: 2
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 72
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 19

Question 20.
Degree: 3
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 73
Answer:

In Exercises 21–28, write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

Question 21.
−5, −1, 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 21

Question 22.
−2, 1, 3
Answer:

Question 23.
3, 4 + i
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 23

Question 24.
2, 5 − i
Answer:

Question 25.
4, −\(\sqrt{5}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 25

Question 26.
3i, 2 − i
Answer:

Question 27.
2, 1 + i, 2 −\(\sqrt{3}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 27

Question 28.
3, 4 + 2i, 1 + \(\sqrt{7}\)
Answer:

ERROR ANALYSIS In Exercises 29 and 30, describe and correct the error in writing a polynomial function with rational coefficients and the given zero(s).

Question 29.
Zeros: 2, 1 + i
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 74
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 29

Question 30.
Zero: 2 +i
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 75
Answer:

Question 31.
OPEN-ENDED Write a polynomial function of degree 6 with zeros 1, 2, and −i. Justify your answer.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 31

Question 32.
REASONING Two zeros of f(x) = x3 − 6x2 − 16x + 96 are 4 and −4. Explain why the third zero must also be a real number.
Answer:

In Exercises 33–40, determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.

Question 33.
g(x) = x4 − x2 − 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 33

Question 34.
g(x) = −x3 + 5x2 + 12
Answer:

Question 35.
g(x) = x3 − 4x2 + 8x + 7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 35

Question 36.
g(x) = x5 − 2x3 − x2 + 6
Answer:

Question 37.
g(x) = x5 − 3x3 + 8x − 10
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 37

Question 38.
g(x) = x5 + 7x4 − 4x3 − 3x2 + 9x − 15
Answer:

Question 39.
g(x) = x6 + x5 − 3x4 + x3 + 5x2 + 9x − 18
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 39

Question 40.
g(x) = x7 + 4x4 − 10x + 25
Answer:

Question 41.
REASONING Which is not a possible classification of zeros for f(x) = x5 − 4x3 + 6x2 + 2x − 6? Explain.
A. three positive real zeros, two negative real zeros, and zero imaginary zeros
B. three positive real zeros, zero negative real zeros, and two imaginary zeros
C. one positive real zero, four negative real zeros, and zero imaginary zeros
D. one positive real zero, two negative real zeros, and two imaginary zeros
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 41

Question 42.
USING STRUCTURE Use Descartes’s Rule of Signs to determine which function has at least 1 positive real zero.
A. f(x) = x4 + 2x3 − 9x2 − 2x – 8
B. f(x) = x4 + 4x3 + 8x2 + 16x + 16
C. f(x) = −x4 − 5x2 − 4
D. f(x) = x4 + 4x3 + 7x2 + 12x + 12
Answer:

Question 43.
MODELING WITH MATHEMATICS From 1890 to 2000, the American Indian, Eskimo, and Aleut population P (in thousands) can be modeled by the function P = 0.004t3 − 0.24t2 + 4.9t + 243, where t is the number of years since 1890. In which year did the population first reach 722,000?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 43

Question 44.
MODELING WITH MATHEMATICS Over a period of 14 years, the number N of inland lakes infested with zebra mussels in a certain state can be modeled by
N = −0.0284t4 + 0.5937t3 − 2.464t2 + 8.33t − 2.5
where t is time (in years). In which year did the number of infested inland lakes first reach 120?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 76
Answer:

Question 45.
MODELING WITH MATHEMATICS For the 12 years that a grocery store has been open, its annual revenue R (in millions of dollars) can be modeled by the function
R = 0.0001(−t4 + 12t3 − 77t2 + 600t + 13,650)where t is the number of years since the store opened. In which year(s) was the revenue $1.5 million?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 77
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 45

Question 46.
MAKING AN ARGUMENT Your friend claims that 2 − i is a complex zero of the polynomial function f(x) = x3 − 2x2 + 2x + 5i, but that its conjugate is not a zero. You claim that both 2 − i and its conjugate must be zeros by the Complex Conjugates Theorem. Who is correct? Justify your answer.
Answer:

Question 47.
MATHEMATICAL CONNECTIONS A solid monument with the dimensions shown is to be built using 1000 cubic feet of marble. What is the value of x?
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 78
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 47

Question 48.
THOUGHT PROVOKING Write and graph a polynomial function of degree 5 that has all positive or negative real zeros. Label each x-intercept. Then write the function in standard form.
Answer:

Question 49.
WRITING The graph of the constant polynomial function f(x) = 2 is a line that does not have any x-intercepts. Does the function contradict the Fundamental Theorem of Algebra? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 49

Question 50.
HOW DO YOU SEE IT? The graph represents a polynomial function of degree 6.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 79
a. How many positive real zeros does the function have? negative real zeros? imaginary zeros?
b. Use Descartes’s Rule of Signs and your answers in part (a) to describe the possible sign changes in the coefficients of f(x).
Answer:

Question 51.
FINDING A PATTERN Use a graphing calculator to graph the function f(x) = (x + 3)n for n = 2, 3, 4, 5, 6, and 7.
a. Compare the graphs when n is even and n is odd.
b. Describe the behavior of the graph near the zero x = −3 as n increases.
c. Use your results from parts (a) and (b) to describe the behavior of the graph of g(x) = (x − 4)20 near x = 4.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 51.1
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 51.2

Question 52.
DRAWING CONCLUSIONS Find the zeros of each function.
f(x) = x5 − 5x + 6
g(x) = x3 − 7x + 6
h(x) = x4 + 2x3 + x2 + 8x − 12
k(x) = x5 − 3x4 − 9x3 + 25x2 − 6x
a. Describe the relationship between the sum of the zeros of a polynomial function and the coefficients of the polynomial function.
b. Describe the relationship between the product of the zeros of a polynomial function and the coefficients of the polynomial function.
Answer:

Question 53.
PROBLEM SOLVING You want to save money so you can buy a used car in four years. At the end of each summer, you deposit $1000 earned from summer jobs into your bank account. The table shows the value of your deposits over the four-year period. In the table, g is the growth factor 1 + r, where r is the annual interest rate expressed as a decimal.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 80
a. Copy and complete the table.
b. Write a polynomial function that gives the value v of your account at the end of the fourth summer in terms of g.
c. You want to buy a car that costs about $4300. What growth factor do you need to obtain this amount? What annual interest rate do you need?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 53.1

Maintaining Mathematical Proficiency

Describe the transformation of f(x) = x2 represented by g. Then graph each function. (Section 2.1)

Question 54.
g(x) = −3x2
Answer:

Question 55.
g(x) = (x − 4)2 + 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 55

Question 56.
g(x) = −(x − 1)2
Answer:

Question 57.
g(x) = 5(x + 4)2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 57

Write a function g whose graph represents the indicated transformation of the graph of f.(Sections 1.2and 2.1)

Question 58.
f(x) = x; vertical shrink by a factor of \(\frac{1}{3}\) and a reflection in the y-axis
Answer:

Question 59.
f(x) = | x + 1 ∣− 3; horizontal stretch by a factor of 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.6 Question 59

Question 60.
f(x) = x2; reflection in the x-axis, followed by a translation 2 units right and 7 units up
Answer:

Lesson 4.7 Tranformations of Polynomial Functions

Essential Question
How can you transform the graph of a polynomial function?

EXPLORATION 1
Transforming the Graph of a Cubic Function
Work with a partner.
The graph of the cubic function f(x) = x3 is shown. The graph of each cubic function g represents a transformation of the graph of f. Write a rule for g. Use a graphing calculator to verify your answers.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 81
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 81.1

EXPLORATION 2
Transforming the Graph of a Quartic Function
Work with a partner.
The graph of the quartic function f(x) = x4 is shown. The graph of each quartic function g represents a transformation of the graph of f. Write a rule for g. Use a graphing calculator to verify your answers.
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 82
Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 82.1

Communicate Your Answer

Question 3.
How can you transform the graph of a polynomial function?

Big Ideas Math Answers Algebra 2 Chapter 4 Polynomial Functions 83
Question 4.

Describe the transformation of f(x) = x4 represented by g(x) = (x + 1)4 + 3. Then graph g.

4.7 Lesson

Monitoring Progress

Question 1.
Describe the transformation of f(x) = x4 represented by g(x) = (x − 3)4 − 1. Then graph each function.

Question 2.
Describe the transformation of f(x) = x3 represented by g(x) = 4(x + 2)3. Then graph each function.

Question 3.
Let f(x) = x5 − 4x + 6 and g(x) = −f(x). Write a rule for g and then graph each function. Describe the graph of g as a transformation of the graph of f.

Question 4.
Let the graph of g be a horizontal stretch by a factor of 2, followed by a translation 3 units to the right of the graph of f(x) = 8x3 + 3. Write a rule for g.

Question 5.
WHAT IF? In Example 5, the height of the pyramid is 6x, and the volume (in cubic feet) is represented by V(x) = 2x3. Write a rule for W. Find and interpret W(7).

Tranformations of Polynomial Functions 4.7 Exercises

Question 1.
COMPLETE THE SENTENCE The graph of f(x) = (x + 2)3 is a ____________ translation of the graph of f(x) = x3.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 1

Question 2.
VOCABULARY Describe how the vertex form of quadratic functions is similar to the form f(x) = a(x − h)3 + k for cubic functions.
Answer:

In Exercises 3–6, describe the transformation of f represented by g. Then graph each function.

Question 3.
f(x) = x4, g(x) = x4 + 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 3

Question 4.
f(x) = x4, g(x) = (x − 5)4
Answer:

Question 5.
f(x) = x5, g(x) = (x − 2)5 − 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 5

Question 6.
f(x) = x6, g(x) = (x + 1)6 − 4
Answer:

ANALYZING RELATIONSHIPS In Exercises 7–10, match the function with the correct transformation of the graph of f. Explain your reasoning.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 84

Question 7.
y = f(x − 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 7

Question 8.
y = f(x + 2) + 2
Answer:

Question 9.
y = f(x − 2) + 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 9

Question 10.
y = f(x) − 2
Answer:

Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 85

In Exercises 11–16, describe the transformation of f represented by g. Then graph each function.

Question 11.
f(x) = x4, g(x) = −2x4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 11

Question 12.
f(x) = x6, g(x) = −3x6
Answer:

Question 13.
f(x) = x3, g(x) = 5x3 + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 13

Question 14.
f(x) = x4, g(x) = \(\frac{1}{2}\)x4 + 1
Answer:

Question 15.
f(x) = x5, g(x) = \(\frac{3}{4}\)(x + 4)5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 15

Question 16.
f(x) = x4, g(x) = (2x)4− 3
Answer:

In Exercises 17–20, write a rule for g and then graph each function. Describe the graph of g as a transformation of the graph of f.

Question 17.
f(x) = x4 + 1, g(x) = f(x + 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 17

Question 18.
f(x) = x5 − 2x + 3, g(x) = 3f(x)
Answer:

Question 19.
f(x) = 2x3 − 2x2 + 6, g(x) = − \(\frac{1}{2}\)f(x)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 19

Question 20.
f(x) = x4 + x3 − 1, g(x) = f(−x) − 5
Answer:

Question 21.
ERROR ANALYSIS Describe and correct the error in graphing the function g(x) = (x + 2)4 − 6
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 86
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 21

Question 22.
ERROR ANALYSIS Describe and correct the error in describing the transformation of the graph of f(x) = x5 represented by the graph of g(x) = (3x)5− 4.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 87
Answer:

Question 23.
f(x) = x3 − 6; translation 3 units left, followed by a reflection in the y-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 23

Question 24.
f(x) = x4 + 2x + 6; vertical stretch by a factor of 2, followed by a translation 4 units right
Answer:

Question 25.
f(x) = x3 + 2x2 − 9; horizontal shrink by a factor of \(\frac{1}{3}\) and a translation 2 units up, followed by a reflection in the x-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 25

Question 26.
f(x) = 2x5 − x3 + x2 + 4; reflection in the y-axis and a vertical stretch by a factor of 3, followed by a translation 1 unit down
Answer:

Question 27.
MODELING WITH MATHEMATICS The volume V(in cubic feet) of the pyramid is given by V(x) = x3 − 4x. The function W(x) = V(3x) gives the volume (in cubic feet) of the pyramid when x is measured in yards. Write a rule for W. Find and interpret W(5).
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 88
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 27

Question 28.
MAKING AN ARGUMENT The volume of a cube with side length x is given by V(x) = x3. Your friend claims that when you divide the volume in half, the volume decreases by a greater amount than when you divide each side length in half. Is your friend correct? Justify your answer.
Answer:

Question 29.
OPEN-ENDED Describe two transformations of the graph of f(x) = x5 where the order in which the transformations are performed is important. Then describe two transformations where the order is not important. Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 29

Question 30.
THOUGHT PROVOKING Write and graph a transformation of the graph of f(x) = x5 − 3x4 + 2x − 4 that results in a graph with a y-intercept of −2.
Answer:

Question 31.
PROBLEM SOLVING A portion of the path that a hummingbird flies while feeding can be modeled by the function
f(x) = −\(\frac{1}{5}\)x(x − 4)2(x − 7), 0 ≤ x ≤ 7 w
here x is the horizontal distance (in meters) and f(x) is the height (in meters). The hummingbird feeds each time it is at ground level.
a. At what distances does the hummingbird feed?
b. A second hummingbird feeds 2 meters farther away than the first hummingbird and flies twice as high. Write a function to model the path of the second hummingbird.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 89
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 31

Question 32.
HOW DO YOU SEE IT? Determine the real zeros of each function. Then describe the transformation of the graph of f that results in the graph of g.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 90
Answer:

Question 33.
MATHEMATICAL CONNECTIONS Write a function V for the volume (in cubic yards) of the right circular cone shown. Then write a function W that gives the volume (in cubic yards) of the cone when x is measured in feet. Find and interpret W(3).
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 91
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 33

Maintaining Mathematical Proficiency

Find the minimum value or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.(Section 2.2)

Question 34.
h(x) = (x + 5)2 − 7
Answer:

Question 35.
f(x) = 4 − x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 35

Question 36.
f(x) = 3(x − 10)(x + 4)
Answer:

Question 37.
g(x) = −(x + 2)(x + 8)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 37

Question 38.
h(x) = \(\frac{1}{2}\)(x − 1)2 − 3
Answer:

Question 39.
f(x) = −2x2 + 4x − 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.7 Question 39

Lesson 4.8 Analyzing Graphs of Polynomial Functions

Essential Question
How many turning points can the graph of a polynomial function have?
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 92
A turning point of the graph of a polynomial function is a point on the graph at which the function changes from

  • increasing to decreasing, or
  • decreasing to increasing.

EXPLORATION 1
Approximating Turning Points
Work with a partner.
Match each polynomial function with its graph. Explain your reasoning. Then use a graphing calculator to approximate the coordinates of the turning points of the graph of the function. Round your answers to the nearest hundredth.
a. f(x) = 2x2 + 3x − 4
b. f(x) = x2 + 3x + 2
c. f(x) = x3 − 2x2 − x + 1
d. f(x) = −x3 + 5x − 2
e. f(x) = x4 − 3x2 + 2x − 1
f. f(x) = −2x5 − x2 + 5x + 3
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 93
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 94

Communicate Your Answer

Question 2.
How many turning points can the graph of a polynomial function have?

Question 3.
Is it possible to sketch the graph of a cubic polynomial function that has no turning points? Justify your answer.

Monitoring Progress

Graph the function.

Question 1.
f(x) = \(\frac{1}{2}\)(x + 1)(x − 4)

Question 2.
f(x) = \(\frac{1}{4}\)(x + 2)(x − 1)(x − 3)

Question 3.
Find all real zeros of f(x) = 18x3 + 21x2 − 13x − 6.

Question 4.
Graph f(x) = 0.5x3 + x2 − x + 2. Identify the x-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing.

Determine whether the function is even, odd, or neither.

Question 5.
f(x) = −x2 + 5

Question 6.
f(x) = x4 − 5x3

Question 7.
f(x) = 2x5

Analyzing Graphs of Polynomial Functions 4.8 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE A local maximum or local minimum of a polynomial function occurs at a ______________ point of the graph of the function.
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 1

Question 2.
WRITING Explain what a local maximum of a function is and how it may be different from the maximum value of the function.
Answer:

ANALYZING RELATIONSHIPS In Exercises 3–6, match the function with its graph.

Question 3.
f(x) = (x − 1)(x − 2)(x + 2)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 3

Question 4.
h(x) = (x + 2)2(x + 1)
Answer:

Question 5.
g(x) = (x + 1)(x − 1)(x + 2)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 5

Question 6.
f(x) = (x − 1)2(x + 2)
Answer:

Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 95

In Exercises 7–14, graph the function.

Question 7.
f(x) = (x − 2)2(x + 1)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 7

Question 8.
f(x) = (x + 2)2(x + 4)2
Answer:

Question 9.
h(x) = (x + 1)2(x − 1)(x − 3)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 9

Question 10.
g(x) = 4(x + 1)(x + 2)(x − 1)
Answer:

Question 11.
h(x) = \(\frac{1}{3}\)(x − 5)(x + 2)(x − 3)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 11

Question 12.
g(x) = \(\frac{1}{12}\)(x + 4)(x + 8)(x − 1)
Answer:

Question 13.
h(x) = (x− 3)(x2 + x + 1)
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 13

Question 14.
f(x) = (x − 4)(2x2 − 2x + 1)
Answer:

ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error in using factors to graph f.

Question 15.
f(x) = (x + 2)(x − 1)2
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 96
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 15

Question 16.
f(x) = x2(x − 3)
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 97
Answer:

In Exercises 17–22, find all real zeros of the function.

Question 17.
f(x) = x3 − 4x2 − x + 4
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 17

Question 18.
f(x) = x3 − 3x2 − 4x + 12
Answer:

Question 19.
h(x) = 2x3 + 7x2 − 5x − 4
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 19

Question 20.
h(x) = 4x3 − 2x2 − 24x − 18
Answer:

Question 21.
g(x) = 4x3 + x2 − 51x + 36
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 21

Question 22.
f(x) = 2x3 − 3x2 − 32x − 15
Answer:

In Exercises 23–30, graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing.

Question 23.
g(x) = 2x3 + 8x2 − 3
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 23

Question 24.
g(x) = −x4 + 3x
Answer:

Question 25.
h(x) = x4 − 3x2 + x
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 25

Question 26.
f(x) = x5 − 4x3 + x2 + 2
Answer:

Question 27.
f(x) = 0.5x3 − 2x + 2.5
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 27

Question 28.
f(x) = 0.7x4 − 3x3 + 5x
Answer:

Question 29.
h(x) = x5 + 2x2 − 17x − 4
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 29

Question 30.
g(x) = x4 − 5x3 + 2x2 + x − 3
Answer:

In Exercises 31–36, estimate the coordinates of each turning point. State whether each corresponds to a local maximum or a local minimum. Then estimate the real zeros and find the least possible degree of the function.

Question 31.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 98
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 31

Question 32.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 99
Answer:

Question 33.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 100
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 33

Question 34.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 101
Answer:

Question 35.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 102
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 35

Question 36.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 103
Answer:

OPEN-ENDED In Exercises 37 and 38, sketch a graph of a polynomial function f having the given characteristics.

Question 37.

  • The graph of f has x-intercepts at x = −4, x = 0, and x = 2.
  • f has a local maximum value when x = 1.
  • f has a local minimum value when x = −2.

Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 37

Question 38.

  • The graph of f has x-intercepts at x = −3, x = 1, and x = 5.
  • f has a local maximum value when x = 1.
  • f has a local minimum value when x = −2 and when x = 4.

Answer:

In Exercises 39–46, determine whether the function is even, odd, or neither.

Question 39.
h(x) = 4x7
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 39

Question 40.
g(x) = −2x6 + x2
Answer:

Question 41.
f(x) = x4 + 3x2 − 2
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 41

Question 42.
f(x) = x5 + 3x3 − x
Answer:

Question 43.
g(x) = x2 + 5x + 1
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 43

Question 44.
f(x) = −x3 + 2x − 9
Answer:

Question 45.
f(x) = x4 − 12x2
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 45

Question 46.
h(x) = x5 + 3x4
Answer:

Question 47.
USING TOOLS When a swimmer does the breaststroke, the function
S = −241t7 + 1060t6 − 1870t5 + 1650t4 − 737t3 + 144t2 − 2.43t
models the speed S (in meters per second) of the swimmer during one complete stroke, where t is the number of seconds since the start of the stroke and 0 ≤ t ≤ 1.22. Use a graphing calculator to graph the function. At what time during the stroke is the swimmer traveling the fastest?
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 104
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 47

Question 48.
USING TOOLS During a recent period of time, the number S (in thousands) of students enrolled in public schools in a certain country can be modeled by S = 1.64x3 − 102x2 + 1710x + 36,300, where x is time (in years). Use a graphing calculator to graph the function for the interval 0 ≤ x ≤ 41. Then describe how the public school enrollment changes over this period of time.
Answer:

Question 49.
WRITING Why is the adjective local, used to describe the maximums and minimums of cubic functions, sometimes not required for quadratic functions?
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 49

Question 50.
HOW DO YOU SEE IT? The graph of a polynomial function is shown.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 105
a. Find the zeros, local maximum, and local minimum values of the function.
b. Compare the x-intercepts of the graphs of y = f(x) and y = −f(x).
c. Compare the maximum and minimum values of the functions y = f(x) and y = −f(x).
Answer:

Question 51.
MAKING AN ARGUMENT Your friend claims that the product of two odd functions is an odd function. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 51

Question 52.
MODELING WITH MATHEMATICS You are making a rectangular box out of a 16-inch-by-20-inch piece of cardboard. The box will be formed by making the cuts shown in the diagram and folding up the sides. You want the box to have the greatest volume possible.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 106
a. How long should you make the cuts?
b. What is the maximum volume?
c. What are the dimensions of the finished box?
Answer:

Question 53.
PROBLEM SOLVING Quonset huts are temporary, all-purpose structures shaped like half-cylinders. You have 1100 square feet of material to build a quonset hut.
a. The surface area S of a quonset hut is given by S = πr2 + πrℓ. Substitute 1100 for S and then write an expression for ℓ in terms of r.
b. The volume V of a quonset hut is given by V = 1 — 2πr2ℓ. Write an equation that gives V as a function in terms of r only.c. Find the value of r that maximizes the volume of the hut.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 107
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 53

Question 54.
THOUGHT PROVOKING Write and graph a polynomial function that has one real zero in each of the intervals −2 < x < −1, 0 < x < 1, and 4 < x < 5. Is there a maximum degree that such a polynomial function can have? Justify your answer.
Answer:

Question 55.
MATHEMATICAL CONNECTIONS A cylinder is inscribed in a sphere of radius 8 inches. Write an equation for the volume of the cylinder as a function of h. Find the value of h that maximizes the volume of the inscribed cylinder. What is the maximum volume of the cylinder?
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 108
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 55.1
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 55.2

Maintaining Mathematical Proficiency

State whether the table displays linear data, quadratic data, or neither. Explain

Question 56.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 109
Answer:

Question 57.
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 110
Answer:
Big Ideas Math Algebra 2 Answer Key Chapter 4 Polynomial Functions 4.8 a 57

Lesson 4.9 Modeling with Polynomial Functions

Essential Question
How can you find a polynomial model for real-life data?

EXPLORATION 1
Modeling Real-Life Data
Work with a partner.
The distance a baseball travels after it is hit depends on the angle at which it was hit and the initial speed. The table shows the distances a baseball hit at an angle of 35° travels at various initial speeds.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 111
a. Recall that when data have equally-spaced x-values, you can analyze patterns in the differences of the y-values to determine what type of function can be used to model the data. If the first differences are constant, then the set of data fits a linear model. If the second differences are constant, then the set of data fits a quadratic model.Find the first and second differences of the data. Are the data linear or quadratic? Explain your reasoning.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 112
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 113
b. Use a graphing calculator to draw a scatter plot of the data. Do the data appear linear or quadratic? Use the regression feature of the graphing calculator to find a linear or quadratic model that best fits the data.12019075400
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 113.1
c. Use the model you found in part (b) to find the distance a baseball travels when it is hit at an angle of 35° and travels at an initial speed of 120 miles per hour.
d. According to the Baseball Almanac, “Any drive over400 feet is noteworthy. A blow of 450 feet shows exceptional power, as the majority of major league players are unable to hit a ball that far. Anything in the 500-foot range is genuinely historic.” Estimate the initial speed of a baseball that travels a distance of 500 feet.

Communicate Your Answer

Question 2.
How can you find a polynomial model for real-life data?

Question 3.
How well does the model you found in Exploration 1(b) fit the data? Do you think the model is valid for any initial speed? Explain your reasoning.

4.9 Lesson

Monitoring Progress

write a cubic function whose graph passes through the given points. 

Question 1.
(−4, 0), (0, 10), (2, 0), (5, 0)

Question 2.
(−1, 0), (0, −12), (2, 0), (3, 0)

Question 3.
Use finite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 118

Use a graphing calculator to find a polynomial function that fits the data.

Question 4.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 119

Question 5.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 120

Modeling with polynomial Functions 4.9 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE When the x-values in a set of data are equally spaced, the differences of consecutive y-values are called ________________.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 1

Question 2.
WRITING Explain how you know when a set of data could be modeled by a cubic function.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, write a cubic function whose graph is shown.

Question 3.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 121
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 3

Question 4.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 122
Answer:

Question 5.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 123
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 5

Question 6.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 124
Answer:

In Exercises 7–12, use finite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function.

Question 7.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 125
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 7

Question 8.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 126
Answer:

Question 9.
(−4, −317), (−3, −37), (−2, 21), (−1, 7), (0, −1), (1, 3), (2, −47), (3, −289), (4, −933)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 9

Question 10.
(−6, 744), (−4, 154), (−2, 4), (0, −6), (2, 16), (4, 154), (6, 684), (8, 2074), (10, 4984)
Answer:

Question 11.
(−2, 968), (−1, 422), (0, 142), (1, 26), (2, −4), (3, −2), (4, 2), (5, 2), (6, 16)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 11

Question 12.
(1, 0), (2, 6), (3, 2), (4, 6), (5, 12), (6, −10), (7, −114), (8, −378), (9, −904)
Answer:

Question 13.
ERROR ANALYSIS Describe and correct the error in writing a cubic function whose graph passes through the given points.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 127
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 13

Question 14.
MODELING WITH MATHEMATICS The dot patterns show pentagonal numbers. The number of dots in the nth pentagonal number is given by f(n) = \(\frac{1}{2}\)n(3n − 1). Show that this function has constant second-order differences.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 128
Answer:

Question 15.
OPEN-ENDED Write three different cubic functions that pass through the points (3, 0), (4, 0), and (2, 6). Justify your answers.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 15

Question 16.
MODELING WITH MATHEMATICS The table shows the ages of cats and their corresponding ages in human years. Find a polynomial model for the data for the first 8 years of a cat’s life. Use the model to estimate the age (in human years) of a cat that is 3 years old.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 129
Answer:

Question 17.
MODELING WITH MATHEMATICS The data in the table show the average speeds y (in miles per hour) of a pontoon boat for several different engine speeds x (in hundreds of revolutions per minute, or RPMs). Find a polynomial model for the data. Estimate the average speed of the pontoon boat when the engine speed is 2800 RPMs.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 130
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 17

Question 18.
HOW DO YOU SEE IT? The graph shows typical speeds y (in feet per second) of a space shuttle x seconds after it is launched.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 131
a. What type of polynomial function models the data? Explain.
b. Which nth-order finite difference should be constant for the function in part (a)? Explain.
Answer:

Question 19.
MATHEMATICAL CONNECTIONS The table shows the number of diagonals for polygons with n sides. Find a polynomial function that fits the data. Determine the total number of diagonals in the decagon shown.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 132
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 133
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 19

Question 20.
MAKING AN ARGUMENT Your friend states that it is not possible to determine the degree of a function given the first-order differences. Is your friend correct? Explain your reasoning.
Answer:

Question 21.
WRITING Explain why you cannot always use finite differences to find a model for real-life data sets.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 21

Question 22.
THOUGHT PROVOKING A, B, and C are zeros of a cubic polynomial function. Choose values for A, B, and C such that the distance from A to B is less than or equal to the distance from A to C. Then write the function using the A, B, and C values you chose.
Answer:

Question 23.
MULTIPLE REPRESENTATIONS Order the polynomial functions according to their degree, from least to greatest.
A.f(x) = −3x + 2x2 + 1
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 134
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 23

Question 24.
ABSTRACT REASONING Substitute the expressions z, z + 1, z + 2, …….. , z + 5 for x in the function f(x) = ax3 + bx2 + cx + d to generate six equally-spaced ordered pairs. Then show that the third-order differences are constant.
Answer:

Maintaining Mathematical Proficiency

Solve the equation using square roots.(Section 3.1)

Question 25.
x2 − 6 = 30
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 25

Question 26.
5x2 − 38 = 187
Answer:

Question 27.
2(x − 3)2 = 24
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 27

Question 28.
\(\frac{4}{3}\)(x + 5)2 = 4
Answer:

Solve the equation using the Quadratic Formula.(Section 3.4)

Question 29.
2x2 + 3x = 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 29

Question 30.
2x2 + \(\frac{1}{2}\) = 2x
Answer:

Question 31.
2x2 + 3x =−3x2 + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 4 Polynomial Functions 4.9 a 31

Question 32.
4x − 20 = x2
Answer:

Polynomial Functions Performance Task: For the Birds-Wildlife Management

Core Vocabulary
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 135

Core Concepts
Section 4.5
Big Ideas Math Answer Key Algebra 2 Chapter 4 Polynomial Functions 136

Section 4.6
Big Ideas Math Answer Key Algebra 2 Chapter 4 Polynomial Functions 137

Section 4.7

Big Ideas Math Answer Key Algebra 2 Chapter 4 Polynomial Functions 138

Section 4.8
Big Ideas Math Answer Key Algebra 2 Chapter 4 Polynomial Functions 139

Section 4.9
Big Ideas Math Answer Key Algebra 2 Chapter 4 Polynomial Functions 140

Mathematical Practices

Question 1.
Explain how understanding the Complex Conjugates Theorem allows you to construct your argument in Exercise 46 on page 203.

Question 2.
Describe how you use structure to accurately match each graph with its transformation in Exercises 7–10 on page 209.

Performance Task
For the Birds -Wildlife Management
How does the presence of humans affect the population of sparrows in a park? Do more humans mean fewer sparrows? Or does the presence of humans increase the number of sparrows up to a point? Are there a minimum number of sparrows that can be found in a park, regardless of how many humans there are? What can a mathematical model tell you?
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 204
To explore the answers to these questions and more, go to BigIdeasMath.com.

Polynomial Functions Chapter Review

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

Question 1.
h(x) = −x3 + 2x2 − 15x7

Question 2.
p(x) = x3 − 5x0.5 + 13x2 + 8

Graph the polynomial function.

Question 3.
h(x) = x2 + 6x5 − 5

Question 4.
f(x) = 3x4 − 5x2 + 1

Question 5.
g(x) = −x4 + x + 2

Find the sum or difference.

Question 6.
(4x3 − 12x2 − 5) − (−8x2 + 4x + 3)

Question 7.
(x4 + 3x3 − x2 + 6) + (2x4 − 3x + 9)

Question 8.
(3x2 + 9x + 13) − (x2 − 2x + 12)

Find the product.

Question 9.
(2y2 + 4y − 7)(y + 3)

Question 10.
(2m + n)3

Question 11.
(s + 2)(s + 4)(s − 3)

Use Pascal’s Triangle to expand the binomial.

Question 12.
(m + 4)4

Question 13.
(3s + 2)5

Question 14.
(z + 1)6

Divide using polynomial long division or synthetic division.

Question 15.
(x3 + x2 + 3x − 4) ÷ (x2 + 2x + 1)

Question 16.
(x4 + 3x3 − 4x2 + 5x + 3) ÷ (x2 + x + 4)

Question 17.
(x4 − x2 − 7) ÷ (x + 4)

Question 18.
Use synthetic division to evaluate g(x) = 4x3 + 2x2 − 4 when x = 5.

Factor the polynomial completely.

Question 19.
64x3 − 8

Question 20.
2z5 − 12z3 + 10z

Question 21.
2a3 − 7a2 − 8a + 28

Question 22.
Show that x + 2 is a factor of f(x) = x4 + 2x3 − 27x − 54. Then factor f(x) completely.

Find all real solutions of the equation.

Question 23.
x3 + 3x2 − 10x − 24 = 0

Question 24.
x3 + 5x2 − 2x − 24 = 0

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

Question 25.
1, 2 − \(\sqrt{3}\)

Question 26.
2, 3, \(\sqrt{5}\)

Question 27.
−2, 5, 3 + \(\sqrt{6}\)

Question 28.
You use 240 cubic inches of clay to make a sculpture shaped as a rectangular prism. The width is 4 inches less than the length and the height is 2 inches more than three times the length. What are the dimensions of the sculpture? Justify your answer.

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

Question 29.
3, 1 + 2i

Question 30.
−1, 2, 4i

Question 31.
−5, −4, 1 −i\(\sqrt{3}\)

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.

Question 32.
f(x) = x4 − 10x + 8

Question 33.
f(x) = −6x4 − x3 + 3x2 + 2x + 18

Describe the transformation of f represented by g. Then graph each function.

Question 34.
f(x) = x3, g(x) = (−x)3 + 2

Question 35.
f(x) = x4, g(x) = −(x + 9)4

Write a rule for g.

Question 36.
Let the graph of g be a horizontal stretch by a factor of 4, followed by a translation 3 units right and 5 units down of the graph of f(x) = x5 + 3x.

Question 37.
Let the graph of g be a translation 5 units up, followed by a reflection in the y-axis of the graph of f(x) = x4 − 2x3 − 12.

Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing.

Question 38.
f(x) = −2x3 − 3x2 − 1

Question 39.
f(x) = x4 + 3x3 − x2 − 8x + 2

Determine whether the function is even, odd, or neither.

Question 40.
f(x) = 2x3 + 3x

Question 41.
g(x) = 3x2 − 7

Question 42.
h(x) = x6 + 3x5

Question 43.
Write a cubic function whose graph passes through the points (−4, 0), (4, 0), (0, 6), and (2, 0).

Question 44.
Use finite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 206

Polynomial Functions Chapter Test

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

Question 1.
3, 1 − \(\sqrt{2}\)

Question 2.
−2, 4, 3i

Find the product or quotient.

Question 3.
(x6 − 4)(x2 − 7x + 5)

Question 4.
(3x4 − 2x3 − x − 1) ÷ (x2 − 2x + 1)

Question 5.
(2x3 − 3x2 + 5x − 1) ÷ (x + 2)

Question 6.
(2x + 3)3

Question 7.
The graphs of f(x) = x4 and g(x) = (x − 3)4 are shown.
a. How many zeros does each function have? Explain.
b. Describe the transformation of f represented by g.
c. Determine the intervals for which the function g is increasing or decreasing.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 207

Question 8.
The volume V (in cubic feet) of an aquarium is modeled by the polynomial function V(x) = x3 + 2x2 − 13x + 10, where x is the length of the tank.
a. Explain how you know x = 4 is not a possible rational zero.
b. Show that x − 1 is a factor of V(x). Then factor V(x) completely.
c. Find the dimensions of the aquarium shown.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 208

Question 9.
One special product pattern is (a − b)2 = a2 − 2ab + b2. Using Pascal’s Triangle to expand (a − b)2 gives 1a2 + 2a(−b) + 1(−b)2. Are the two expressions equivalent? Explain.

Question 10.
Can you use the synthetic division procedure that you learned in this chapter to divide any two polynomials? Explain.

Question 11.
Let T be the number (in thousands) of new truck sales. Let C be the number (in thousands) of new car sales. During a 10-year period, T and C can be modeled by the following equations where t is time (in years).
T = 23t4 − 330t3 + 3500t2 − 7500t + 9000
C = 14t4 − 330t3 + 2400t2 − 5900t + 8900
a. Find a new model S for the total number of new vehicle sales.
b. Is the function S even, odd, or neither? Explain your reasoning.

Question 12.
Your friend has started a golf caddy business. The table shows the profits p (in dollars) of the business in the first 5 months. Use finite differences to find a polynomial model for the data. Then use the model to predict the profit after 7 months.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 209

Polynomial Functions Cumulative Assessment

Question 1.
The synthetic division below represents f(x) ÷ (x− 3). Choose a value for m so that x − 3 is a factor of f(x). Justify your answer.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 210

Question 2.
Analyze the graph of the polynomial function to determine the sign of the leading coefficient, the degree of the function, and the number of real zeros. Explain.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 211

Question 3.
Which statement about the graph of the equation 12(x− 6) = −( y + 4)2 is not true?
A. The vertex is (6, −4).
B. The axis of symmetry is y = −4.
C. The focus is (3, −4).
D. The graph represents a function.

Question 4.
A parabola passes through the point shown in the graph. The equation of the axis of symmetry is x = −a. Which of the given points could lie on the parabola? If the axis of symmetry was x = a, then which points could lie on the parabola? Explain your reasoning.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 212

Question 5.
Select values for the function to model each transformation of the graph of f(x) = x.
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 213
a. The graph is a translation 2 units up and 3 units left.
b. The graph is a translation 2 units right and 3 units down.
c. The graph is a vertical stretch by a factor of 2, followed by a translation 2 units up.
d. The graph is a translation 3 units right and a vertical shrink by a factor of \(\frac{1}{2}\), followed by a translation 4 units down.

Question 6.
The diagram shows a circle inscribed in a square. The area of the shaded region is21.5 square meters. To the nearest tenth of a meter, how long is each side of the square?
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 214
A. 4.6 meters
B. 8.7 meters
C. 9.7 meters
D. 10.0 meters

Question 7.
Classify each function as even, odd, or neither. Justify your answer.
a. f(x) = 3x5
b. f(x) = 4x3 + 8x
c. f(x) = 3x5 + 12x2+ 1
d. f(x) = 2x4
e. f(x) = x11 − x7
f. f(x) = 2x8 + 4x4 + x2 − 5

Question 8.
The volume of the rectangular prism shown is given by V = 2x3 + 7x2 − 18x − 63. Which polynomial represents the area of the base of the prism?
A. 2x2 + x − 21
B. 2x2 + 21 − x
C. 13x + 21 + 2x2
D. 2x2 − 21 − 13x
Big Ideas Math Algebra 2 Solutions Chapter 4 Polynomial Functions 215

Question 9.
The number R (in tens of thousands) of retirees receiving Social Security benefits is represented by the function R = 0.286t3 − 4.68t2 + 8.8t + 403, 0 ≤ t ≤ 10
where t represents the number of years since 2000. Identify any turning points on the given interval. What does a turning point represent in this situation?

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