Factoring a polynomial of degree 4

Question

I am attempting to factor a relatively benign looking polynomial of degree 4. I have tried to use synthetic division to factor this. I was hoping to be able to get a remainder of zero at some point. I don't quite know how this works but I suspect that if perhaps I extend things to complex numbers I might have some luck. For this same reason, I am suspecting that all of the roots are may be complex. Can someone tell me how one deals with this?

Answer 1

In general, something like synthetic division along with the Rational Root Theorem might not be the worst idea, assuming there is a real root. Otherwise, it might be a product of two irreducible quadratic polynomials in which case you may want to look at something like factor-by-grouping. In the worst scenario, you have an 'exotic' polynomial and will have to solve for the roots. Luckily, mathematicians in the 1500s managed to find formulas - albeit of the headache inducing variety - to do exactly this, see the Wiki page for quartic functions.

Answer 2

Using a CAS can greatly reduce the headache.