Show that if $w$ is a zero of $P(z)$ (a complex polynomial) than $1/w$ is also

Question

Let $P(z) = 2z^4-5z^3+4z^2-5z+2.$

if $w$ is a zero of $P(z)$ show that $1/w$ is also a zero of that polynomial.

Answer 1

Just note that $z^4P(\dfrac1z)=P(z)$.

Answer 2

We must substitute $$w->\frac{1}{w}$$ in your equation: $$\frac{2}{w^4}-\frac{5}{w^3}+\frac{4}{w^2}-\frac{5}{w}+2$$ and this must be zero if $$2w^4-5w^3+4w^2-5w+2=0$$