Question

How could you use Descartes' Rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial, as well as find the number of possible positive and negative real roots to a polynomial?

Answer 1

`\bf f(x)=x^5\stackrel{\downarrow }{-}x^4\stackrel{\downarrow }{+}x^3\stackrel{\downarrow }{-}x^2\stackrel{\downarrow }{+}5\qquad \impliedby \textit{4 sign changes}`

so the number of positive real roots is either 4, or (4-2) 2, or (2-2) 0. And the negative real roots are only 1. Any slack gets picked up by the versatile complex twins.

4 real positive ones and 1 negative real one

or

2 real positive ones and 1 negative real one and 2 complex ones

or

0 real positive ones and 1 negative real one and 4 complex ones.