In each of the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.$f(x) = x^3 - 6x^2 + 11x - 6, g(x) = x^2 - 3x + 2$

Given:

$f(x) = x^3 - 6x^2 + 11x - 6, g(x) = x^2 - 3x + 2$

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.

$f(x) = x^3 - 6x^2 + 11x - 6$

$g(x) = x^2 - 3x + 2$

$x^2-2x-x+2=0$

$x(x-2)-1(x-2)=0$

$(x-1)(x-2)=0$

$x-1=0$ or $x-2=0$

$x=1$ or $x=2$

So, the remainders will be $f(1)$ and $f(2)$.

$f(1) = (1)^3-6(1)^2 +11(1)-6$

$= 1-6+11-6$

$=12-12$

$=0$

$f(2)=(2)^3-6(2)^2 +11(2)-6$

$= 8-6(4)+22-6$

$=30-30$

$=0$

This implies, $(x-1)$ and $(x-2)$ are factors of $f(x)$

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