Factor Theorem
Factor theorem is mainly used to factor the polynomials and to find the n roots of the polynomials. Factor theorem is very helpful for analyzing polynomial equations. In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
What is Factor Theorem?
Factor theorem is a special kind of the polynomial remainder theorem that links the factors of a polynomial and its zeros. The factor theorem removes all the known zeros from a given polynomial equation and leaves all the unknown zeros. The resultant polynomial has a lower degree in which the zeros can be easily found out.
Zero of a Polynomial
Before learning about the factor theorem, it is essential for us to know about the zero or a root of the polynomial. We say that y = a is a root or zero of a polynomial, g(y) only if 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. Let's consider an example to find the zeros of the seconddegree polynomial g(y) = y^{2} + 2y − 15. To do this we simply solve the equation by using the factorization of quadratic equation method as:
y^{2} + 2y − 15
= (y+5)(y−3)
= 0
⇒ y =−5 and y = 3
Thus, this seconddegree polynomial y^{2} + 2y − 15 has two zeros or roots which are 5 and 3.
Factor Theorem Formula
As per the factor 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. Here, a is any real number. The formula of the factor theorem is g(y) = (y – a) q(y). It is important to note that all the following statements apply for any polynomial g(y):
 (y – a) is a factor of g(y).
 g(a) = 0.
 The remainder becomes 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.
Proof of Factor Theorem
In order to prove the factor theorem, let's first consider a polynomial g(y) that is being divided by (y – a) only if g(a) = 0. By using the division algorithm, the given polynomial can be written as the product of its divisor and its quotient:
Dividend = (Divisor × Quotient ) + Remainder
⟹ g(y) = (y – a) q(y) + remainder. Here, 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 substitute g(a) = 0 then the remainder is 0,
⟹ g(y) = (y – a) q(y) + 0
⟹ g(y) = (y – a) q(y)
Thus, we can say that (y – a) is a factor of the polynomial g(y). Here we can see that the factor theorem is actually 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 i.e., g(a) = 0.
How to Use the Factor Theorem?
Let's learn how to use the factor theorem with an example. Check whether (y + 5) is a factor of 2y^{2} + 7y – 15 or not. Given that, y + 5 = 0. Then, y = 5. Now let's 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
= 0
Thus, y + 5 is a factor of 2y^{2} + 7y – 15.
Using the Factor Theorem To Factor a ThirdDegree Polynomial
We usually use the factorization method to factor seconddegree or quadratic polynomials. For higher degrees, we can use the belowgiven procedure to factor the polynomial:
 Step 1: Use the synthetic division of the polynomial method to divide the given polynomial g(y) by the given binomial (y−a)
 Step 2: After the completion of the division, confirm whether the remainder is 0. If the remainder is not zero, then it means that (ya) is not a factor of g(y).
 Step 3: Using the division algorithm, write the given polynomial as the product of (ya) and the quadratic quotient q(y)
 Step 4: If it is possible, factor the quadratic quotient further.
 Step 5: Express the given polynomial as the product of its factors.
Using the factor theorem, let's show that (y+2) is a factor of y^{3} − 6y^{2} − y + 30 and then find the remaining factors. After finding the remaining factors, we will use these factors to determine the zeros of the given polynomial.
 The first step is to use the synthetic division method to show that (y+2) is a factor of the thirddegree polynomial y^{3} − 6y^{2} − y + 30.
 After the completion of the synthetic division method, we find that the remainder is zero. Hence, (y+2) is a factor of the given polynomial.
 Now, let's use the division algorithm to write the given polynomial as the product of the divisor (y+2) and the quadratic quotient (y^{2}8y+15) as (y+2) (y^{2}−8y+15).
 Let's factorize the quadratic equation to write the polynomial as (y+2)(y−3)(y−5).
Thus, by using the factor theorem, the zeros of the given polynomial y^{3} − 6y^{2} − y + 30 are –2, 3, and 5.
Difference Between Factor Theorem and Remainder Theorem
Factor theorem and remainder theorem are similar but they refer to two different concepts. The remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros. Let's consider an example of a polynomial g(y) = y^{2} − 2y + 1 to understand the difference:
For the given polynomial g(y), let's use the remainder theorem and put 3 as y into g(y):
g(3) = (3)^{2} − 2(3) + 1
g(3) = 9 − 6 + 1
g(3) = 4
Hence, by the remainder theorem, the remainder when we divide y^{2} − 2y + 1 by y−3 is 4. We can apply this in reverse too by dividing y^{2} − 2y + 1 by y−3, and we will get the remainder equal to the value of g(3).
For the same polynomial, let's use the factor theorem g(y) = y^{2} − 2y + 1 equals 0 when y =1.
g(1) = (1)^{2} − 2(1) + 1
g(1) = 1 − 2 + 1
g(1) = 0
This tells us that (y−1) is a factor of y^{2} − 2y + 1. We can also apply this in reverse. We can factor y^{2} − 2y + 1 into (y−1)^{2}. Thus, 1 is a zero of g(y).
Related Articles on Factor Theorem
Also, check the following pages related to the factor theorem.
 Zeros of Quadratic Polynomial
 Factorization of Algebraic Expressions
 Polynomial Functions
 Degree of a Polynomial
 Types of Polynomials
Important Notes on Factor Theorem
Here is a list of a few points that should be remembered while studying the factor theorem:
 Factor theorem is mainly used to factor the polynomials and to find the n roots of that polynomial.
 In real life, factoring is useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
 As per the factor 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.
Factor Theorem Examples

Example 1: Use the factor theorem to check whether y + 1 is a factor of the polynomial 3y^{4} + y^{3} – y^{2} + 3y + 2, or not.
Solution:
Given y + 1; then, y + 1 = 0, we get y = 1.
Substitute y = 1 in the given polynomial equation 3y^{4} + y^{3} – y^{2} + 3y + 2.
⟹ 3(–1)^{4} + (–1)^{3} – (–1)^{2} +3(–1) + 2
= 3(1) + (–1) – 1 – 3 + 2= 3 1 1 3 + 2
Adding all positive terms and all negative terms, we get,
= 5  5
= 0
Therefore, we can say that y + 1 is a factor of 3y^{4} + y^{3} – y^{2} + 3y + 2. 
Example 2: Check whether 2y + 1 is a factor of the polynomial 4y^{3} + 4y^{2} – y – 1 or not using the factor theorem.
Solution:
Let's equate the given binomial 2y + 1 = 0.
∴ y = 1/2
Substitute y = 1/2 in the given polynomial equation 4y^{3} + 4y^{2} – y – 1.
⟹ 4( 1/2)^{3} + 4(1/2)^{2} – (1/2) – 1
= 1/2 + 1 + 1/2 – 1
= 0
The remainder = 0, thus, 2y + 1 is a factor of the polynomial equation 4y^{3} + 4y^{2} – y – 1.
FAQs on Factor Theorem
How do you Use the Factor Theorem?
We can use the belowgiven procedure to factor the polynomial using factor theorem:
 Step 1: Use the synthetic division of the polynomial method to divide the given polynomial g(y) by the given binomial (y−a)
 Step 2: After the completion of the division, confirm whether the remainder is 0. If the remainder is not zero, then it means that (ya) is not a factor of g(y).
 Step 3: Using the division algorithm, write the given polynomial as the product of (ya) and the quadratic quotient q(y)
 Step 4: If it is possible, factor the quadratic quotient further by splitting the middle term method
 Step 5: Express the given polynomial as the product of its factors.
What is the Factor Theorem Formula?
As per the factor 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. Here, a is any real number. The formula of the factor theorem is g(y) = (y – a) q(y). It is important to note that all the following statements apply for any polynomial g(y):
 (y – a) is a factor of g(y).
 g(a) = 0.
 The remainder becomes 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.
Explain Factor Theorem With an Example.
Let's use the factor theorem to find whether y+2 is a factor of the polynomial g(y) = y^{3} + 3y^{2} + 5y + 6 or not.
g(–2) = (–2)^{3} + 3(–2)^{2} + 5(–2) + 6
g(2) = –8 + 12 – 10 + 6
g(2) = 0
This tells us that (y+2) is a factor of y^{3} + 3y^{2} + 5y + 6. We can also apply this in reverse.
What is the Importance of the Factor Theorem?
Factor theorem is mainly used to factor the polynomials and to find the n roots of that polynomial. It is a special kind of the polynomial remainder theorem that links the factors of a polynomial and its zeros. The factor theorem removes all the known zeros from a given polynomial equation and leaves all the unknown zeros. The resultant polynomial has a lower degree in which the zeros are a lot easier to find.
What is the Difference Between Factor Theorem and Remainder Theorem?
The difference between the factor theorem and the remainder theorem is that the remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros.
Where do we Use the Factor Theorem in Real Life?
In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
What is the Importance of the Remainder Theorem and Factor Theorem?
Both the factor theorem and the remainder theorem come in handy to find the factors of a polynomial without using the other methods like synthetic division, long division, or any other traditional methods of factoring.
Is Factor Theorem and Remainder Theorem the Same?
No, the factor theorem and remainder theorem are not the same. While the remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros.