Question

By using factor theorem in the following examples, determine whether q ( x ) is a factor p ( x ) or not.

(i) $p\left(x\right)=x^3-x^2-x-1,q\left(x\right)=x-1$

(ii) $p\left(x\right)=2x^3-x^2-45,q\left(x\right)=x-3$

Answer 1

(i)
$p\left(x\right)=x^3-x^2-x-1$

Divisor = $q\left(x\right)=x-1$

$\therefore p\left(1\right)={\left(1\right)}^3-{\left(1\right)}^2-1-1=1-1-1-1=-2$

Since p(1) ≠ 0, so by factor theorem $q\left(x\right)=x-1$ is not a factor of polynomial $p\left(x\right)=x^3-x^2-x-1$.

(ii)
$p\left(x\right)=2x^3-x^2-45$

Divisor = $q\left(x\right)=x-3$

$\therefore p\left(3\right)=2\times {\left(3\right)}^3-{\left(3\right)}^2-45=2\times 27-9-45=54-54=0$

So, by factor theorem $q\left(x\right)=x-3$ is a factor of polynomial $p\left(x\right)=2x^3-x^2-45$.