In mathematics, the factor theorem establishes a relationship between the factors and zeros of a polynomial. It is a unique case consideration of the polynomial remainder theorem, Factor theorem is mainly used to factor the polynomials and to find the n-roots of the polynomials. It is very helpful for analyzing the polynomial equations. In our daily life, factoring can be useful when exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
The Factor theorem is a special case consideration of the polynomial remainder theorem. It is frequently linked with the remainder theorem, therefore do not confuse both. On this platform, we will discuss the 10th Grade Math Concept factor theorem, Statement, formula, proof, how to use factor theorem, example problems, and so on.
What is Factor Theorem?
Factor theorem is a type of the polynomial remainder theorem that links the factors of a polynomial and its zeros. It will remove all the known zeros from a given polynomial equation and leaves all the unknown zeros. The resultant polynomial will have a lower degree in which the zeros can be easily found.
Factor Theorem Statement
The factor theorem states that, if f(x) is a polynomial of degree ‘n’ greater than or equal to 1, and ‘a’ is any real number, then (x – a) is a factor of f(x) if f(a) = 0. In other terms, we say that (x – a) is a factor of f(x) if f(a) = 0.
Zero of a Polynomial:
It is essential for us to know about the zero or a root of the polynomial, before learning about the factor theorem. We say that y = a is a root or zero of a polynomial g(y) only when g(a) = 0. We can also say that y = a is a root or zero of a polynomial only if it is a solution to the equation g(y) = 0. Consider an example to find the zeros of the second-degree polynomial is g(y) = y2 + 2y − 15.
You can do this we can solve the equation by using the factorization of the quadratic equation method as:
y2 + 2y − 15 = (y+5)(y−3)
Equate the factors to zero, then it will be
y =−5 and y = 3
Thus, the second-degree polynomial y2 + 2y − 15 has two zeros or roots which are – 5 and 3.
Factor Theorem Formula
As per the theorem, (y – a) can be considered as a factor of the polynomial g(y) of degree n ≥ 1, if and only if g(a) = 0. In this, a is any real number. So, the formula of the factor theorem is g(y) = (y – a) q(y). It is very important to note that all the following statements apply to any polynomial g(y):
- (y–a) is a factor of g(y).
- g(a) = 0.
- The remainder will become zero when g(y) is divided by (y – a).
- The solution to g(y) = 0 is a and the zero of the function g(y) is a.
Factor Theorem Proof
In order to prove the factor theorem, first, consider a polynomial g(y) that is being divided by (y – a) only when g(a) = 0. While using the division algorithm, we can write the given polynomial as the product of its divisor and its quotient. So, it will be
Dividend = (Divisor × Quotient ) + Remainder.
i.e., g(y) = (y – a) q(y) + remainder.
Where g(y) is the dividend, (y – a) is the divisor, and q(y) is the quotient. From the remainder theorem, we get
g(y) = (y – a) q(y) + g(a)
If we substituted g(a) = 0 then the remainder is 0.
g(y) = (y – a) q(y) + 0
g(y) = (y – a) q(y)
Thus, we will say that (y – a) is a factor of the polynomial g(y). Here we can see that the factor theorem is a result of the remainder theorem, which states that a polynomial g(y) has a factor (y – a), if and only if, a is a root that is g(a) = 0.
How to Use the Factor Theorem?
Now, learn how to use the factor theorem with an example. The steps are below-given to find the factors of a polynomial using factor theorem:
- Step 1 : If f(-c)=0, then (x+ c) is a factor of the polynomial f(x).
- Step 2 : If p(d/c)= 0, then (cx-d) is a factor of the polynomial f(x).
- Step 3 : If p(-d/c)= 0, then (cx+d) is a factor of the polynomial f(x).
- Step 4 : If p(c)=0 and p(d) =0, then (x-c) and (x-d) are factors of the polynomial p(x).
Example: Check whether (y + 5) is a factor of 2y2 + 7y – 15 or not?
Solution: Given that, y + 5 = 0.
First, equate y+5=0 that is y = – 5.
Next, substitute y =- 5 into the given polynomial equation. We get,
g(-5) = 2(-5)2+ 7(-5) – 15
= 2 (25) – 35 – 15
= 50 – 35 – 15 = 50-50= 0.
Hence, the y+5 is a factor of 2y2 + 7y – 15.
Use the Factor Theorem to Find Factors of a Third-Degree Polynomial
Generally, we use the factorization method to find the factors of second-degree or quadratic polynomials. For higher degrees or more than two degrees, we can use the following steps to find factors of the polynomial:
- Step 1: Use the synthetic division of the polynomial method, you will divide the given polynomial g(y) by the given binomial (y−a).
- Step 2: After the completion of the division, first confirm whether the remainder is 0 or not. If the remainder is not zero, then (y-a) is not a factor of g(y).
- Step 3: Using the division algorithm, we will write the given polynomial as the product of (y-a) and the quadratic quotient q(y).
- Step 4: If it is possible, you will factor the quadratic quotient further.
- Step 5: Next, Express the given polynomial as the product of its factors.
Example: Using the factor theorem, show that (y+2) is a factor of y3− 6y2 − y + 30 and then find the remaining factors. So, after finding the remaining factors, we will use these factors to determine the zeros of the given polynomial.
The step-by-step procedure for finding the factors of the given polynomial.
Step 1: The first step is to use the synthetic division method to show that (y+2) is a factor of the third-degree polynomial y3− 6y2 − y + 30.
Step 2: After completing the synthetic division method, we find that the remainder is zero. Hence, (y + 2) is a factor of the given polynomial.
Step 3: Now, let us use the division algorithm to write the given polynomial as the product of the divisor (y + 2) and the quadratic quotient (y2– 8y +15), that is, y3 − 6y2 − y + 30 = (y+2) (y2− 8y +15).
Step 4: Let’s factorize the quadratic equation y2− 8y +15 to write the polynomial as (y + 2)(y − 3)(y − 5).
Thus, by using the factor theorem, the zeros of the given polynomial y3 − 6y2 − y + 30 are –2, 3, and 5.
Factor Theorem Questions and Answers
Problem 1: For a curve that crosses the x-axis at 3 points, of which one is at 2. What is the factor of 2x3−x2−7x+2?
Given that, the polynomial expression is 2x3−x2−7x+2.
Now, we need to find the factors of the given polynomial.
First, find the degree of the polynomial.
The polynomial for the equation degree is 3 and could be all easy to solve.
Substitute x value as ‘2’ in the given polynomial, then it will be
f(2) = 2(2)3−(2)2−7(2)+2.
Now, this will be, 16−4−14+2
=12 (-14) +2 = -2 + 2 = 0
So, f(2) =0, we have found a factor and a root.
Therefore, (x-2) should be a factor of 2x3−x2−7x+2.
Problem 2: Find out whether x + 1 is a factor of the below-given polynomial.
i) 3x4 + x3–x2+3x+2
As given in the question,
The polynomial expression is, f(x) = 3x4 + x3–x2+3x+2
The factor is x+1.
Now, we need to check whether the given factor is right or wrong.
Using the factor theorem,
Let f(x) = 3x4 + x3–x2+3x+2.
Equate the factor to zero. We get,
x = -1.
After substituting x value in the polynomial. We get,
f (–1) = 3(-1)4 + (-1)3–(-1)2+3(-1)+2
i.e., 3(1) + (–1) – 1 – 3 + 2 = 0.
Yes, x+1 is the right factor.
Hence, the (x + 1) is a factor of the given function f(x).
FAQ’s on Factor Theorem
1. What are the other methods to find the factors of polynomials?
Apart from the factor theorem, we can use the polynomial long division method and synthetic division method to find the factors of the polynomial.
2. Explain the formula of the factor theorem?
The Factor Theorem explains to us that if the remainder f(r) = R = 0, then (x − r) happens to be a factor of f(x). The Factor Theorem is quite important because of its usefulness to find the roots of polynomial equations.
3. Is it possible for a remainder to be negative?
No, a remainder can never be a negative number.
4. What is the importance of the Factor Theorem?
The importance of the factor theorem is,
1. Factor theorem is mainly used to factor the polynomials and to find the n roots of that polynomial.
2. In real life, factoring is useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
3. As per the theorem, the (y – a) can be considered as a factor of the polynomial g(y) of degree n ≥ 1, if and only if g(a) = 0.