Let P(z)=anzn+an−1zn−1+⋯+a1z+a0 be a polynomial of degree n.
(a) Suppose that an,an−1,…,a1,a0 are all real. Show that, if z1 is a root of P, then z1 is also a root. In other words, the roots must be complex conjugates, something you likely learned without proof in high school.
(b) Find an example of a polynomial that has some roots occurring as complex conjugates, and some not.