# Descartes' Rule of Signs Help

• Jan 17th 2008, 09:20 AM
Descartes' Rule of Signs Help
I need help figuring out this problem:

Use Descartes' Rule of Signs to analyze the number of positive and negative real roots and the number of non-real roots of the function:
http://hosted.webwork.rochester.edu/...04137e56c1.png
There are at least __ and at most ___ positive real roots.
There are at least___ and at most___ negative real roots.
There are at least ___and at most ____ non-real roots.
• Jan 17th 2008, 09:45 AM
CaptainBlack
Quote:

I need help figuring out this problem:

Use Descartes' Rule of Signs to analyze the number of positive and negative real roots and the number of non-real roots of the function:
http://hosted.webwork.rochester.edu/...04137e56c1.png
There are at least __ and at most ___ positive real roots.
There are at least___ and at most___ negative real roots.
There are at least ___and at most ____ non-real roots.

The coeficients of $h(x)$ change sign $3$ times, hence there are at most $3$
positive roots of $h(x)$. As the number of positive roots can only change by
multiples of $2$ the minimum number of real roots is $1$.

The coefficients of $h(-x)$ change sign $6$ times, thus there are at most $6$
negative roots of $h(x)$ , and the minimum number is $0$.

As $3+6=9$ and there are a total of $9$ real or non-real roots there may be as
few as $0$ non-real roots. The maximum number is $8$ (when there is one real
root).

RonL