Question

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

f(x) = $3x^3+x^2$ - 20x + 12, g(x) = 3x - 2

Answer 1

f(x) = $3x^3+x^2$ - 20x + 12, g(x) = 3x - 2

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

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

$=3{\left(\frac{2}{3}\right)}^3+{\left(\frac{2}{3}\right)}^2-20\times \frac{2}{3}+12$

= $3\times \frac{8}{27}+\frac{4}{9}-\frac{40}{3}+12$

= $\frac{8}{9}+\frac{4}{9}-\frac{40}{3}+\frac{12}{1}$

=$\frac{8+4-120+108}{9}=\frac{120-120}{9}$

=$\frac{0}{9}=0$

$\because$ Remainder = 0,

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