In each of the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.$f(x) = 2x^3 - 9x^2 + x + 12, g(x) = 3 - 2x$

Given:

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

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.

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

$g(x) = 3 - 2x$

$3-2x=0$

$3=2x$

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

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

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

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

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

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

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

$=\frac{81-81}{4}$

$=0$

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

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Updated on: 10-Oct-2022

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