Using factor theorem, factorize each of the following polynomials:$x^3 + 2x^2 - x - 2$

Given:

Given expression is $x^3 + 2x^2 - x - 2$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=x^{3}+2 x^{2}-x-2$

The factors of the constant term $-2$ are $\pm 1, \pm 2$.

Let $x=1$, this implies,

$f(1)=(1)^{3}+2(1)^{2}-(1)-2$

$=1+2-1-2$

$=3-3$

$=0$

Therefore $x-1$ is a factor of $f(x)$

Let $x=-1$, this implies,

$f(-1)=(-1)^{3}+2(-1)^{2}-(-1)-2$

$=-1+2+1-2$

$=3-3$

$=0$

Therefore $x+1$ is a factor of $f(x)$.

Let $x=2$, this implies,

$f(-2)=(-2)^{3}+2(-2)^{2}-(-2)-2$

$=-8+8+2-2$

$=0$

Therefore $(x+2)$ is a factor of $f(x)$.

Hence, $f(x)=(x+1)(x-1)(x+2)$.