Using factor theorem, factorize each of the following polynomials:$x^3 + 13x^2 + 32x + 20$

Given:

Given expression is $x^3 + 13x^2 + 32x + 20$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=x^{3}+13 x^{2}+32 x+20$

The factors of the constant term 20 are $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$.

Let $x=-1$, this implies,

$f(-1)=(-1)^{3}+13(-1)^{2}+32(-1)+20$

$=-1+13-32+20$

$=33-33$

$=0$

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

Let $x=-2$, this implies,

$f(-2)=(-2)^{3}+13(2)^{2}+32(-2)+20$

$=-8+52-64+20$

$=72-72$

$=0$

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

Dividing $f(x)=x^{3}+13 x^{2}+32 x+20$ by $(x+1)(x+2)=x^{2}+3 x+2$, we have,

$x^{2}+3 x+2$) $x^{3}+13 x^{2}+32 x+20$($x+10$

                            $x^{3}+3 x^{2}+2 x$

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

                                       $10 x^{2}+30 x+20$
                                       $10 x^{2}+30 x+20$

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

                                                       0

Hence, $x^{3}+13 x^{2}+32 x+20=(x+1)(x+2)(x+10)$.

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Updated on: 10-Oct-2022

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