Provide an appropriate response.Why does a function defined by a polynomial of degree five with real coefficients have either 1, 3, or 5 real zeros counting multiplicities?

Question

What will be an ideal response?

Answer 1

By the Fundamental Theorem of Algebra and the Factor Theorem, the polynomial has five complex, not necessarily real, zeros. By the Conjugate Zeros Theorem, the nonreal zeros among these come in pairs, so there are 0, 2, or 4 nonreal zeros. That leaves 5, 3, or 1 real zeros.