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.

Factor Theorem Statement

The factor theorem states that If $p(x)$ is a polynomial of degree $n\geq 1$ and $a$ is any real number, then

• $x-a$ is a factor of $p(x)$, if $p(a)=0$
• $p(a)=0$, if $x-a$ is a factor of $p(x)$

Example 6 : Examine whether $x+2$ is a factor of $x^{3}+3x^{2}+5x+6$ and of $2x+4$.

Solution : The zero of $x+2$ is $-2$.

Let $p(x)=x^{3}+3x^{2}+5x+6$

$p(-2)=(-2)^{3}+3(-2)^{2}+5(-2)+6$

$p(-2)=-8+3(4)-10+6=0$

$s(x)=2x+4$

$s(-2)=2(-2)+4=0$

Hence $x+2$ is a factor of $x^{3}+3x^{2}+5x+6$ and of $2x+4$.

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

$g(-5)=2(-5)^{2}+7(-5)-15$

$g(-5)=2(25)+-35-15$

$g(-5)=50+-35-15=0$

Hence, $y+5$ is a factor of $2y^{2}+7y-15$