# Eureka Math Algebra 2 Module 1 Lesson 4 Answer Key

## Engage NY Eureka Math Algebra 2 Module 1 Lesson 4 Answer Key

### Eureka Math Algebra 2 Module 1 Lesson 4 Example Answer Key

Example 1.
If x = 10, then the division 1573 ÷ 13 can be represented using polynomial division. The quotient is x2 + 2x + 1
The completed board work for this example should look something like this: Example 2.
Use the long division algorithm for polynomials to evaluate
$$\frac{2 x^{3}-4 x^{2}+2}{2 x-2}$$
The quotient is x2 – 2x – 1

### Eureka Math Algebra 2 Module 1 Lesson 4 Opening Exercise Answer Key

Exercise 1.
Use the reverse tabular method to determine the quotient $$\frac{2 x^{3}+11 x^{2}+7 x+10}{x+5}$$ Exercise 2.
Use your work from Exercise 1 to write the polynomial 2x3 + 11x2 + 7x + 10 in factored form, and then multiply the factors to check your work above.
(x + 5) (2x2 + x + 2) The product is 2x3 + 11x2 + 7x + 10

### Eureka Math Algebra 2 Module 1 Lesson 4 Exercise Answer Key

Use the long division algorithm to determine the quotient. For each problem, check your work by using the reverse tabular method.

Exercise 1.
$$\frac{x^{2}+6 x+9}{x+3}$$
x + 3

Exercise 2.
$$\frac{7 x^{3}-8 x^{2}-13 x+2}{7 x-1}$$
x2 – x – 2

Exercise 3.
$$\frac{x^{3}-27}{x-3}$$
x2 + 3x + 9

Exercise 4.
$$\frac{2 x^{4}+14 x^{3}+x^{2}-21 x-6}{2 x^{2}-3}$$
x2 + 7x + 2

Exercise 5.
$$\frac{5 x^{4}-6 x^{2}+1}{x^{2}-1}$$
5x2 – 1

Exercise 6.
$$\frac{x^{6}+4 x^{4}-4 x-1}{x^{3}-1}$$
x3 + 4x + 1

Exercise 7.
$$\frac{2 x^{7}+x^{5}-4 x^{3}+14 x^{2}-2 x+7}{2 x^{2}+1}$$
x5 – 2x + 7

Exercise 8.
$$\frac{x^{6}-64}{x+2}$$
x5 – 2x4 + 4x3 – 8x2 + 16x – 32

### Eureka Math Algebra 2 Module 1 Lesson 4 Problem Set Answer Key

Use the long division algorithm to determine the quotient in problems 1 – 5.

Question 1.
$$\frac{2 x^{3}-13 x^{2}-x+3}{2 x+1}$$
x2 – 7x + 3

Question 2.
$$\frac{3 x^{3}+4 x^{2}+7 x+22}{x+2}$$
3x2 – 2x + 11

Question 3.
$$\frac{x^{4}+6 x^{3}-7 x^{2}-24 x+12}{x^{2}-4}$$
x2 + 6x – 3

Question 4.
(12x4 + 2x3 + x – 3) ÷ (2x2 + 1)
6x2 + x – 3

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

Question 6.
Use long division to find the polynomial, p, that satisfies the equation below.
2x4 – 3x2 – 2 = (2x2 + 1) (p(x))
p(x) = x2 – 2

Question 7.
Given q(x) = 3x3 – 4x2 + 5x + k
a. Determine the value of k so that 3x – 7 is a factor of the polynomial
k = – 28

b. What is the quotient when you divide the polynomial q by 3x – 7?
x2 + x + 4

Question 8.
In parts (a) – (b) and (d) – (e), use long division to evaluate each quotient. Then, answer the remaining questions.
a. $$\frac{x^{2}-9}{x+3}$$
x – 3

b. $$\frac{x^{4}-81}{x+3}$$
x3 – 3x2 + 9x – 27

c. Is x + 3 a factor of x3 – 27 Explain your answer using the long division algorithm.
No. The remainder is not when you perform long division.

d. $$\frac{x^{3}+27}{x+3}$$
x2 – 3x + 9

e. $$\frac{x^{5}+243}{x+3}$$
x4 – 3x3 + 9x2 – 27x + 81

f. Is x + 3 a factor of x2 + 9 Explain your answer using the long division algorithm.
No. The remainder is not 0 when you perform long division.

g. For which positive integers n is x + 3 a factor of xn + 3n? Explain your reasoning.
Only if n is an odd number. By extending the patterns in parts (a) – (c) and (e), we can generalize that x + 3 divides evenly into xn + 3n for odd powers of n only.

h. If is a positive integer, is x + 3 a factor of xn + 3n Explain your reasoning.
$$\frac{2 x^{2}-3 x-5}{x+1}$$