MATH QUESTION!! (help)?

Question

A polynomial function of degree n with real coefficients has exactly n complex zeros. At most n of them are real zeros.

true or false? and why?

thanks so much

Answer 1

(Assuming it's a polynomial in one variable, with a nonzero leading coefficient, and n>0)

The statements are true. The first is true as a consequence of the fundamental theorem of algebra, which is actually a bit broader; it says (quoting cited source #1)

Every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if repeated roots are counted up to their multiplicity.

Since the set of real numbers is a subset of the set of complex numbers, the above statement includes polynomials with real coefficients. And since each such polynomial of degree n has exactly n complex zeros, clearly no more than n of the zeros are real. There can, of course, be fewer than n real zeros (including no real zeros, if n is even).

If the coefficients are real, any complex zeros must appear in complex-conjugate pairs.