Factorize each of the following polynomials:$x^3 + 13x^2 + 31x - 45$ given that $x + 9$ is a factor.

Given:

Given expression is $x^3 + 13x^2 + 31x - 45$ and $x + 9$ is a factor.

To do:

We have to factorize the given polynomial.

Solution:

Let $f(x)=x^{3}+13 x^{2}+31 x-45$

Dividing $f(x)$ by $x+9$, we get,

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

                $x^3+9x^2$

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

                           $4x^2+31x-45$

                           $4x^2+36x$

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

                                      $-5x-45$

                                     $-5x-45$

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

                                            0

$f(x)=(x+9)(x^{2}+4 x-5)$

$=(x+9)(x^{2}+5 x-x-5)$

$=(x+9)[x(x+5)-1(x+5)]$

$=(x+9)(x+5)(x-1)$

Hence, $x^3 + 13x^2 + 31x - 45=(x+9)(x+5)(x-1)$.

Tutorialspoint

Updated on: 10-Oct-2022

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