# 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?

Jan 14, 2016

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 $3$ different times.

This means that there are $3$ positive roots. There could also be $1$--I'll return to this idea later.

To find the negative roots, count the sign changes in $f \left(- x\right)$:

$f \left(- x\right) = - {x}^{5} - 4 {x}^{4} - 3 {x}^{3} - 2 x - 6$

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 $3$ positive roots. Since there are no negative roots, the other $2$ roots (we know there are $5$ since the degree of the polynomial is $5$) are complex roots.

However, since complex roots always come in pairs, the $3$ positive roots may actually be expressed as $1$ positive root and $2$ complex roots. Again, there are the two additional complex roots not a part of the $3$, so the polynomial could have $1$ positive root and $4$ complex ones.

We now know that we only have to try to find the positive roots. The potential roots are $1 , 2 , 3 , 6$. Synthetically divide or do polynomial long division to find that $\left(x - 3\right)$ is a root.

$\frac{{x}^{5} - 4 {x}^{4} + 3 {x}^{3} + 2 x - 6}{x - 3} = {x}^{4} - {x}^{3} + 2$

The four imaginary roots are remarkably complex (this is just one): $x = \frac{1}{4} - \frac{1}{2} \sqrt{\frac{1}{4} + {\left(9 + i \sqrt{1455}\right)}^{\frac{1}{3}} / {3}^{\frac{2}{3}} + \frac{8}{3 \left(9 + i \sqrt{1455}\right)} ^ \left(\frac{1}{3}\right)} - \frac{1}{2} \sqrt{\frac{1}{2} - {\left(9 + i \sqrt{1455}\right)}^{\frac{1}{3}} / {3}^{\frac{2}{3}} - \frac{8}{3 \left(9 + i \sqrt{1455}\right)} ^ \left(\frac{1}{3}\right) - \frac{1}{4 \sqrt{\frac{1}{4} + {\left(9 + i \sqrt{1455}\right)}^{\frac{1}{3}} / {3}^{\frac{2}{3}} + \frac{8}{3 \left(9 + i \sqrt{1455}\right)} ^ \left(\frac{1}{3}\right)}}}$

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]}