Question

Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x).

f(x) = $2x^3-9x^2$ + x + 12, g(x) = 3 - 2x

Answer 1

f(x) = $2x^3-9x^2$ + x + 12, g(x) = 3 - 2x

Let 3 - 2x = 0, then 2x = 3 $\Rightarrow x=\frac{3}{2}$

Now, Remainder $=f\left(\frac{3}{2}\right)$

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

$=2\times \frac{27}{8}-9\times \frac{9}{4}+\frac{3}{2}+12$

$=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+\frac{12}{1}$

$=\frac{27-81+6+48}{4}=\frac{81-81}{4}=\frac{0}{4}=0$

$\therefore$ Remainder = 0

$\therefore$ (3-2x) is a factor of f(x)