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Using factor theorem, factorize each of the following polynomials:$x^3 + 2x^2 - x - 2$
Given:
Given expression is $x^3 + 2x^2 - x - 2$.
To do:
We have to find the given polynomial using factor theorem.
Solution:
Let $f(x)=x^{3}+2 x^{2}-x-2$
The factors of the constant term $-2$ are $\pm 1, \pm 2$.
Let $x=1$, this implies,
$f(1)=(1)^{3}+2(1)^{2}-(1)-2$
$=1+2-1-2$
$=3-3$
$=0$
Therefore $x-1$ is a factor of $f(x)$
Let $x=-1$, this implies,
$f(-1)=(-1)^{3}+2(-1)^{2}-(-1)-2$
$=-1+2+1-2$
$=3-3$
$=0$
Therefore $x+1$ is a factor of $f(x)$.
Let $x=2$, this implies,
$f(-2)=(-2)^{3}+2(-2)^{2}-(-2)-2$
$=-8+8+2-2$
$=0$
Therefore $(x+2)$ is a factor of $f(x)$.
Hence, $f(x)=(x+1)(x-1)(x+2)$.
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