Question

Use the Factor Theorem to determine whether $g\left(x\right)$ is factor of $f\left(x\right)$ in the following case:
$f\left(x\right)=4x^3+20x^2+33x+18,g\left(x\right)=2x+3$

Answer 1

Given $f\left(x\right)=4x^3+20x^2+33x+18$ and $g\left(x\right)=2x+3$

Put $2x+3=0$

$⟹x=-\frac{3}{2}$

$2x+3$ is a factor of $f\left(x\right)$ if $f\left(\frac{-3}{2}\right)=0$

replace $x$ by $-\frac{3}{2}$ in $f\left(x\right)$ we get,

$f(-\frac{3}{2})=4{(-\frac{3}{2})}^3+20{(-\frac{3}{2})}^2+33(-\frac{3}{2})+18$

$⟹f(-\frac{3}{2})=-4\times \frac{27}{8}+20\times \frac{9}{4}-\frac{99}{2}+18$

$⟹f(-\frac{3}{2})=-\frac{-54}{4}+\frac{180}{4}-\frac{99}{2}+18$

$⟹f(-\frac{3}{2})=\frac{-54+180-198+72}{4}⟹f(-\frac{3}{2})=0$

Therefore, $g\left(x\right)$ is the factor of $f\left(x\right)=4x^3+20x^2+33x+18$