Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ be a polynomial of degree n
Suppose not all of $a_n,a_{n-1},\ldots,a_1,a_0$ are real. Show that $P$ has at least one root whose complex conjugate is not a root.
This problem suggests me to prove the contrapositive. Any advice?
(What I understand for contrapositive: "If the roots of $P$ are complex conjugates, then all of $a_n,a_{n-1},\ldots,a_1,a_0$ are real.")
If so, I believe I should prove that $(z-z_1)(z-\bar {z_1})$ is a polynomial with real coefficients (I know how to do this) and that multiplication of polynomials with real coefficients is a polynomial with real coefficients.
Correct me if I'm wrong.