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