How do I factor a polynomial function with a degree higher than 2 without guessing numbers of $\frac{p}{q}$?

Question

I have an equation $f(x)=x^4+4x^3+2x^22-x+6$. In the past I was taught to factor it by getting the zeros by getting $p/q$, and start guessing zeros, and plugging them into the function. Once I got one or two, I would try to divide the function by them to get the rest.

It would seem to me that there has to be a much easier way of doing this. Some kind of trick. If the equation was only something like $f(x)=x^2+5x-6$, then it would be easy. just find the number that multiplies together to equal $-6$, and adds up to equal $5$. The answer would be $(x+6)(x-1)$. Is there some trick like this for functions with a degree higher than $2$?

Answer 1

The corresponding "trick" is to just find the four numbers $a,b,c,d$ such that $a+b+c+d=-4$, $ab+ac+ad+bc+bd+cd=2$, $abc+abd+acd+bcd=1$, and $abcd=6$.

Answer 2

It rather depends on what you mean by guessing zeros.

The rational root theorem provides a systematic algorithm for finding rational roots of the form $\frac{p}{q}$: $p$ divides the coefficient of the constant (so is $\pm 1, \pm 2, \pm 3 \text{ or } \pm 6$ in your example), while $q$ divides the coefficient of the highest power (so is $\pm 1$ in your example).

You then test all of the possible pairs (or decide just to stick to positive $q$ to save half the effort) and in your example find that the only rational roots are $-\frac{2}{1}$ and $-\frac{3}{1}$. Then factor these out and check that none of them are repeated zeros.