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

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 $f(x)$

Dividing $f(x)$ by $x-1$, we have,

$x-1$) $x^{3}- 2x^{2}-x+2$($x^2-x-2$

             $x^{3}-x^{2}$

         ----------------------------

                     $-x^{2}-x+2$                      

                     $-x^{2}+x$

                 --------------------------

                                   $-2x+2$

                                   $-2x+2$

                              -----------------

                                        0               

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

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

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

Hence, $x^{3}-2 x^{2}- x+2=(x-1)(x+1)(x-2)$.