Given $f(X)$ of degree $n$, guess that $a$ is a zero of $f(X)$ and check $f(a)=0$; then write

where $g(X)$ has degree $n-1$; then guess that $b$ is a zero of $g(X)$ and check $g(b)=0$; then

where $h(X)$ has degree $n-2$; and so on until we find all the zeros we can.