Do every polynomial function has at least one complex zero?

Answer 1

No. Complex zeros always come in conjugate pairs. So if a+bi is one zero, then a-bi is also a zero.

The fundamental theorem of algebra says

"Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers."

If you want to know how many complex root a given polynomial has, you might consider finding out how many real roots it has. This can be done with Descartes Rules of signs

• The maximum number of positive real roots can be found by counting the number of sign changes in f(x). The actual number of positive real roots may be the maximum, or the maximum decreased by a multiple of two.
• The maximum number of negative real roots can be found by counting the number of sign changes in f(-x). The actual number of negative real roots may be the maximum, or the maximum decreased by a multiple of two.
• Complex roots always come in pairs. That's why the number of positive or number of negative roots must decrease by two. Using the two rules for positive and negative signs along with the fact that complex roots come in pairs, you can determine the number of complex roots.