How do you determine the number of complex roots of a polynomial of degree n?
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra (FTOA) tells us that any non-constant polynomial in one variable with Complex (possibly Real) coefficients has a zero in
A straightforward corollary of this (often stated as part of the FTOA) is that a polynomial of degree
So a simple answer to your question would be that a polynomial of degree
How many of those
If the polynomial has Real coefficients, then any Complex zeros will occur in Complex conjugate pairs. So the number of non-Real zeros will be even.
Descartes' Rule of Signs
If the coefficients are Real then we can find out some more things about the zeros by looking at the signs of the coefficients.
To determine the possible number of negative Real zeros, look at the signs of the coefficients of
For example, consider:
#f(x) = x^4+x^3-x^2+x-2#
The signs of the coefficients are in the pattern
Since there are
#f(-x) = x^4-x^3-x^2-x-2#
has coefficients with signs
Since there is
Since the total number of zeros of
Discriminants are another useful tool, which I will describe in another answer.