# How do you determine the number of complex roots of a polynomial of degree n?

##### 1 Answer

#### Answer:

See explanation...

#### Explanation:

**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.

If

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**

Discriminants are another useful tool, which I will describe in another answer.