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

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

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

RonL