Question

In the following case, use factor theorem to find whether $g\left(x\right)$ is a factor of the polynomial $p\left(x\right)$ or not.
$p\left(x\right)=x^3-3x^2+6x-20g\left(x\right)=x-2$

Answer 1

The Factor Theorem states that if $a$ is the root of any polynomial $p\left(x\right)$ that is if $p\left(a\right)=0$, then $(x-a)$ is the factor of the polynomial $p\left(x\right)$.

It is given that $p\left(x\right)=x^3-3x^2+6x-20$ and $g\left(x\right)=x-2$, therefore, by factor theorem $(x-2)$ is the factor of $p\left(x\right)$ if $p\left(2\right)=0$ and thus,

$p\left(2\right)=2^3-(3\times 2^2)+(6\times 2)-20=8-(3\times 4)+12-20=8-12+12-20=-12\ne 0$

Since $p\left(2\right)\ne 0$

Hence, $(x-2)$ is not a factor of $p\left(x\right)=x^3-3x^2+6x-20$.