Finding factors of polynomials without every power descending

Question

There is a way for finding factors of polynomials with the biggest power being n in $a^n$, and other powers descending from $n-1$ in $b^{n-1}$ to $0$ in for example $z_0$. And it is solved by finding $p$ and $q$, and putting every possible $p/q$. What if we don't have all the powers? My example could be $$4x^4 - 5x^2 +6x -8$$ Please give a solution so at least we don't have to try out every number, and it is reduced to a finite amount of numbers.

Answer 1

The rational root theorem does work in examples like yours. Note that the $x^3$ term isn't really missing. It's just hidden away. Your polynomial is equal to $$ 4x^4 +0x^3-5x^2 + 6x-8 $$ and nothing in the rational root theorem says it stops working if some of the coefficients are $0$ (just that the first and last coefficients must be non-zero).