# Thread: Descartes' Rule of Signs Help

1. ## 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:

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.

2. Originally Posted by badandy328
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:

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