# Factor theorem of polynomial

In this chapter we will discuss the concept of factor theorem of polynomial and will also solve some questions related to it.

## What is factor of polynomial ?

The expression which completely divides the polynomial without leaving any remainder is called factor of polynomial.

For example, consider the following polynomial f(x).

f(x) = \mathtt{x^{2} +2x+1}

Here (x + 1) is the factor of above polynomial f(x) as if we divide f(x) by (x + 1), the expression will be fully divisible.

Let’s check the division.

\mathtt{\Longrightarrow \ \frac{\mathtt{x^{2} +2x+1}}{x+1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x+1)^{2}}{x+1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{( x+1)}( x+1)}{\cancel{x+1}}}

You can see that the denominator completely divides numerator without leaving any remainder.

Hence, (x + 1) is factor of polynomial f(x).

## What is factor theorem of polynomial ?

Let us consider two polynomial f(x) and g(x).

If we put root of g(x) on f(x) and get 0 as a result, it means that g(x) is the factor of polynomial f(x).

Let us understand the theorem with the help of examples.

Question 01
Consider the two below polynomials.

f(x) = \mathtt{x^{3} +3x^{2} +5x+6}

g(x) = x + 2

Check if g(x) is the factor of polynomial f(x).

Solution
We will solve the question using factor theorem of polynomials.

Lets first find the root of g(x).

x + 2 = 0

x = -2

Hence, -2 is the root of function g(x).
Put this root on polynomial f(x). If we get 0 then g(x) is the factor of f(x).

\mathtt{\Longrightarrow \ ( -2)^{3} +3( -2)^{2} +5( -2) +6\ }\\\ \\ \mathtt{\Longrightarrow \ \ -8+12-10+6}\\\ \\ \mathtt{\Longrightarrow \ 18\ -\ 18}\\\ \\ \mathtt{\Longrightarrow \ 0}

Hence. g(x) is the factor of f(x).

Question 02
Consider the two polynomials f(x) and g(x).

f(x)= \mathtt{x^{3} \ –\ 4x^{2} \ +\ x\ +\ 6}

g(x) = x – 3

Check if g(x) is the factor of f(x).

Solution
Let’s first find the root of g(x)

x – 3 = 0

x = 3

Put this value of x in polynomial f(x).

\mathtt{\Longrightarrow \ 3^{3} \ –\ 4( 3)^{2} \ +\ 3\ +\ 6}\\\ \\ \mathtt{\Longrightarrow \ 27-36+\ 9}\\\ \\ \mathtt{\Longrightarrow \ 0}

Since we get 0, it means that g(x) is the factor of polynomial f(x).

Question 03
Consider the two functions f(x) and g(x).

f(x) = \mathtt{2x^{3} +x^{2} \ –\ 2x\ –\ 1}

g(x) = x +1

Check if g(x) is the factor of polynomial f(x).

Solution
Let’s first find the root of g(x).

x + 1 = 0

x = -1

Put this value of x in polynomial f(x).

\mathtt{\Longrightarrow \ 2( -1)^{3} +( -1)^{2} \ –\ 2( -1) \ –\ 1}\\\ \\ \mathtt{\Longrightarrow \ -2+1+2-1}\\\ \\ \mathtt{\Longrightarrow \ 0}

Since we are getting 0, it tells that g(x) is factor of f(x).

Next chapter : Problems on factorization of polynomials

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