Question
Using factor theorem, factorize the following polynomials:
$x^3 + 2x^2 - x - 2$
$x^3 + 2x^2 - x - 2$
✓
Answer
Let $x=1 f(1)=1^3+2(1)^2-1-2=0$
$\therefore x=1$ is a solution
$\Rightarrow x -1=0$ i. e $( x -1)$ is a factor of $f ( x )$
By division algorithm $x^3+2 x^2-x-2$
$=(x-1)\left(x^2+3 x+2\right)$
$=(x-1)\left(x^2+2 x+x+2\right)$
$=(x-1)(x(x+2)+1(x+2))$
$=(x-1)(x+2)(x+1)$
$\therefore x^3+2 x^2-x-2=(x-1)(x+2)(x+1)$
$\therefore x=1$ is a solution
$\Rightarrow x -1=0$ i. e $( x -1)$ is a factor of $f ( x )$
By division algorithm $x^3+2 x^2-x-2$
$=(x-1)\left(x^2+3 x+2\right)$
$=(x-1)\left(x^2+2 x+x+2\right)$
$=(x-1)(x(x+2)+1(x+2))$
$=(x-1)(x+2)(x+1)$
$\therefore x^3+2 x^2-x-2=(x-1)(x+2)(x+1)$
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