Using factor theorem, factorize each of the following polynomials:$x^3 - 3x^2 - 9x - 5$

Given:

Given expression is $x^3 - 3x^2 - 9x - 5$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=x^{3}-3 x^{2}-9 x-5$.

The factors of the constant term $-5$ are $\pm 1, \pm 5$

Let $x=-1$, this implies,

$f(-1)=(-1)^{3}-3(-1)^{2}-9(-1)-5$

$=-1-3+9-5$

$=9-9$

$=0$

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

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

$x+1$) $x^{3}- 3x^{2}-9 x-5$($x^2-4x-5$

             $x^{3}+x^{2}$

         ----------------------------

                     $-4x^{2}-9 x-5$
                     $-4x^{2}-4x$

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

                                   $-5x-5$

                                   $-5x-5$

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

                                        0               

$x^2-4x-5=x^2+x-5x-5$

$=x(x+1)-5(x+1)$

$=(x+1)(x-5)$

Hence, $x^{3}-3 x^{2}-9 x-5=(x+1)(x+1)(x-5)$.