Question

(X-2) is a factor of x^4+2x^3-7x^2-8x+12. The other factors are ____, ____, and _____

Answer 1

We have

$\dfrac{x^4+2x^3-7x^2-8x+12}{x-2}=x^3+4x^2+x-6$

The rational root theorem suggests that other possible roots may be -6, 6, -3, 3, -2, 2, -1, and 1. It turns out that $x=-2$ is a root, since $(-2)^3+4(-2)^2+(-2)-6=0$, so $x+2$ is also a factor and we have

$\dfrac{x^4+2x^3-7x^2-8x+12}{(x-2)(x+2)}=x^2+2x-3$

Finally, we can factorize the remaining quotient easily:

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

so the other factors are $x+2$, $x+3$, and $x-1$.