In each of the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.$f(x) = 3x^3 + x^2 - 20x + 12, g(x) = 3x - 2$

Given:

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

To do:

We have to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.

Solution:

We know that, if $g(x)$ is a factor of $f(x)$, then the remainder will be zero.

$f(x) = 3x^3  + x^2 - 20x + 12$

$g(x) = 3x - 2$

$3x-2=0$

$3x=2$

$x=\frac{2}{3}$

So, the remainder will be $f(\frac{2}{3})$.

$f(\frac{2}{3}) = 3(\frac{2}{3})^3+(\frac{2}{3})^2 -20(\frac{2}{3})+12$

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

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

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

$=\frac{12-120+108}{9}$

$=\frac{120-120}{9}$

$=0$

Therefore, $g(x)$ is a factor of polynomial $f(x)$.    

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Updated on: 10-Oct-2022

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