How do you factor this? $x^3 + x - 2$

Question

How do you factor $x^3 + x - 2$?

Hint: Write it as $(x^3-x^2+x^2-x+2x-2)$ to get $(x-1)(x^2+x+2)$

Note the factored form here. Thanks!

Answer 1

By inspection, we see that $1$ is a root of $x^3 + x - 2$, so it is divisible by $x - 1$; alternatively, the rational roots theorem would suggest this too.

Now $x^2 + x + 2 = x^2 + x + \frac{1}{4} + \frac{7}{4} = (x + \frac{1}{2})^2 + \frac{7}{4}$ has no real roots, and is irreducible. If you're factoring over $\Bbb{C}$, then it's got roots at $\pm \sqrt{\frac 7 4}i - \frac{1}{2}$; denoting these as $\alpha_+$ and $\alpha_-$, the original polynomial then splits as

$$(x - 1)(x - \alpha_+)(x - \alpha_-)$$

Answer 2

Note that the summation of the coefficients is $0$: $$+1+1+(-2)=0$$ so the polynomial has a factor like $(x-1)$.