What is $|x^3-x^2-x+1|$?

Question

What is $|x^3 - x^2 -x +1|$?

I have calculated that $$|x^3 - x^2 -x +1| = \begin{cases} -x^3 +x^2 +x -1,\ x <-1\\\\ x^3 - x^2 -x +1, x \geq -1 \end{cases}$$

Am I right?

Answer 1

Yes it's true, you can see it directly by seeing that $$x^3-x^2-x+1=(x+1)(x-1)^2.$$

Answer 2

$P(x)=x^3 - x^2 -x +1=x^2(x-1)-(x-1)=(x-1)(x^2-1)=(x-1)^2(x+1)$

Since $(x-1)^2\geq 0$, sign of $P(x)$ depends on $x+1$

If $x<-1$ then $P(x)<0$ and $|x^3 - x^2 -x +1|=-x^3 + x^2 + x - 1$

If $x\geq-1$ then $P(x)\geq 0$ and $|x^3 - x^2 -x +1|=x^3 - x^2 - x + 1$