# How do you find all the real and complex roots and use Descartes Rule of Signs to analyze the zeros of #P(x) = x^5 - 4x^4 + 3x^3 + 2x - 6#?

##### 1 Answer

First, count the number of sign changes in the polynomial, from term to term.

Using P(ositive) and N(egative), it goes: P N P P N

Thus, the sign changes

This means that there are

To find the negative roots, count the sign changes in

The signs go: N N N N N

The sign never changes so there are no negative roots.

So, we know that there are a possible

However, since complex roots always come in pairs, the

We now know that we only have to try to find the *positive roots*. The potential roots are

The four imaginary roots are remarkably complex (this is just one):

There *is* a method to find the roots to a quartic equation, but it is very convoluted.

Check a graph:

graph{x^5 - 4x^4 + 3x^3 + 2x - 6 [-29.71, 28.03, -18.82, 10.05]}