How would you go about finding the roots of the polynomial:
$$x^4 +5x^3+4x^2+6x-4=0.$$
I attempt to form two quadratics e.g.
$$(x^2+ax+b)(x^2+cx+d)$$
and then tried to expand, collect like terms, and solve for the coefficients simultaneously, but this results in a hard system of equations.
I was wondering if there was a better approach to this question.
Any help would be greatly appreciated.
Best Answer
Note that $x^4-4=(x^2+2)(x^2-2)$. This suggests that you might try to express your polynomial as$$(x^2+ax+2)(x^2+bx-2)=x^4+(a+b)x^3+abx^2+2(-a+b)x-4.$$It is now easy to see that all it takes is to choose $a=1$ and $b=4$.