How can I find the roots of the polynomial $12x^{4}+2x^3+10x^2+2x-2$?

Question

It's clear that I can divide by $2$, but I don't know what can I do with $$6x^{4}+x^3+5x^2+x-1$$

Is there any algorithm for it or a trick? I have found the roots by an online calculator but I don't know how can I calculate them. Thank you for your help.

Answer 1

Here, I try to give a way of factorization, which isn't too hard to be noticed:
$6x^4+x^3+5x^2+x-1$
$=5x^4+x^3+5x^2+x+x^4-1$
$=x^3(5x+1)+x(5x+1)+(x^2+1)(x^2-1)$
$=x(x^2+1)(5x+1)+(x^2+1)(x^2-1)$
$=(x^2+1)(6x^2+x-1)$
$=(x^2+1)(3x-1)(2x+1)$

Answer 2

The hint.

Easy to see that $i$ is a root, which gives a factor $x^2+1.$

Answer 3

Another hint: In general if you search for rational roots and try inserting $x=p/q$ (irreducible), then $6p^4+p^3 q+ 5 p^2q^2+p q^3 -q^4=0$ implies that $q$ should divide $6$ and $p$ should divide 1. For details look up Rational root theorem in wikipedia.

In the present situation you will find $1/3$, $-1/2$ in this way. If you include the possibility of $p$ being imaginary then you also pick up $\pm i$ (but this is perhaps a bit cheating).