Using a computer we can see that the only real root of $f(y)=-2y^5 +4y^4-2y^3-y=0$ is $0$. Furthermore, we know from algebra that since this polynomial lives in $\Bbb R[y]$ that the roots come in complex conjugate pairs. I.e. we knew that there were either $1,3$ or $5$ real roots of this polynomial. Furthermore, if we could guess factors, we could complete polynomial long division to break up the polynomial. Noting that $y=0$ is a root, we want the $4$ roots of:
- How would we deduce that the remaining roots are not real?
- How would we find these by hand.