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**Mathstud28** Descartes rule of signs: Let $\displaystyle p(x)$ be defined as before. The amount of sign changes in $\displaystyle p(x)$ is the number of real, positive roots of $\displaystyle p(x)$. The number of sign changes in $\displaystyle p(-x)$ is the number of real, negative roots of $\displaystyle p(x)$.

So let's combine these two rules. As before let the polynomial $\displaystyle p(x)$ be of degree $\displaystyle n$. Let there be $\displaystyle m$ sign changes in $\displaystyle p(x)$ and $\displaystyle e$ sign changes in $\displaystyle p(-x)$. By the FTA we must have $\displaystyle n$ roots exactly, so the amount of roots left is $\displaystyle n-m-e=i$. This number $\displaystyle i$ is the number of imaginary roots of $\displaystyle p(x)$