Fundamental Theorem of Algebra

Every polynomial equation with complex coordinates and a degree greater than zero has at least one root in the set of complex numbers.

A polynomial equation with degree n will have n roots in the set of complex numbers.

Descartes’ Rule of Signs

Descartes’ Rule of Signs can be used to determine the number of positive real zeros, negative real zeros, and imaginary zeros in a polynomial function.

How many zeros (and what kinds of zeros) does this equation have?

After arranging the terms of a polynomial equation into descending powers:

The number of positive real zeros in y = P(x) is equal to the number of changes of sign in front of each term, or is less than this by an even number

and

The number of negative real zeros in y = P(x) is the same as the number of changes of sign in front of the terms of P(-x), or is less than this value by an even number.

First, we test for the number of positive real zeros:

Second, we test for the number of negative real zeros:

So, how many different combinations of zeros (and what kinds of zeros) does this equation have?