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