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Using factor theorem, factorize each of the following polynomials:$x^3 - 3x^2 - 9x - 5$
Given:
Given expression is $x^3 - 3x^2 - 9x - 5$.
To do:
We have to find the given polynomial using factor theorem.
Solution:
Let $f(x)=x^{3}-3 x^{2}-9 x-5$.
The factors of the constant term $-5$ are $\pm 1, \pm 5$
Let $x=-1$, this implies,
$f(-1)=(-1)^{3}-3(-1)^{2}-9(-1)-5$
$=-1-3+9-5$
$=9-9$
$=0$
Therefore, $x+1$ is a factor of $f(x)$
Dividing $f(x)$ by $x+1$, we have,
$x+1$) $x^{3}- 3x^{2}-9 x-5$($x^2-4x-5$
$x^{3}+x^{2}$
----------------------------
$-4x^{2}-9 x-5$
$-4x^{2}-4x$
--------------------------
$-5x-5$
$-5x-5$
-----------------
0
$x^2-4x-5=x^2+x-5x-5$
$=x(x+1)-5(x+1)$
$=(x+1)(x-5)$
Hence, $x^{3}-3 x^{2}-9 x-5=(x+1)(x+1)(x-5)$.
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