Using factor theorem, factorize each of the following polynomials:$x^3 -10x^2 - 53x - 42$


Given:

Given expression is $x^3 -10x^2 - 53x - 42$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=x^{3}-10 x^{2}-53 x-42$.

The factors of the constant term $-42$ are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42$
Let $x=-1$, this implies,

$f(-1)=(-1)^{3}-10(-1)^{2}-53(-1)-42$

$=-1-10+53-42$

$=53-53$

$=0$

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

Let $x=-3$, this implies,

$f(-3)=(-3)^{3}-10(-3)^{2}-53(-3)-42$

$=-27-90+159-42$

$=159-159$

$=0$

Therefore, $x+3$ is a factor of $f(x)$

Dividing $f(x)$ by $(x+1)(x+3)=x^2+4x+3$, we have,

$x^{2}+4 x+3$) $x^{3}-10 x^{2}-53 x-42$($x-14$

                            $x^{3}+4 x^{2}+3 x$

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

                                      $-14 x^{2}-56 x-42$

                                      $-14 x^{2}-56 x-42$

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

                                                 0

Therefore, $x^{3}-10^{2}-53 x-42=(x+1)(x+3)(x-14)$. 

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Updated on: 10-Oct-2022

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