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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 $f(x)=x^{3}-2x^{2}-x+2$.
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 $f(x)$
Dividing $f(x)$ by $x-1$, we have,
$x-1$) $x^{3}- 2x^{2}-x+2$($x^2-x-2$
$x^{3}-x^{2}$
----------------------------
$-x^{2}-x+2$
$-x^{2}+x$
--------------------------
$-2x+2$
$-2x+2$
-----------------
0
$x^2-x-2=x^2-2x+x-2$
$=x(x-2)+1(x-2)$
$=(x+1)(x-2)$
Hence, $x^{3}-2 x^{2}- x+2=(x-1)(x+1)(x-2)$.
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