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