Question

Show that g(x) is a factor of f(x), where

f(x) = 2x³ − 9x² + x + 12, g(x) = 3 − 2x

Answer 1

It is given that $f\left(x\right)=2x^3-9x^2+x+12$ and $g\left(x\right)=(3-2x)$

By factor theorem, (3 − 2x) is the factor of f(x), if $f\left(\frac{3}{2}\right)=0$

Therefore,

In order to prove that (3 − 2x) is a factor of f(x). It is sufficient to show that

$f\left(\frac{3}{2}\right)=0$

Now,

$f\left(\frac{3}{2}\right)=2{\left(\frac{3}{2}\right)}^3-9{\left(\frac{3}{2}\right)}^3+\left(\frac{3}{2}\right)+12$

$=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12$

$=\frac{27-81+6}{4}+12$

$=\frac{-48}{4}+12$

$=\frac{-48+48}{4}$

$=0$

Hence, (3 − 2x), is the factor of polynomial f(x).