#3**+2 **

The wording of this question strongly suggests that the use of Descartes' Rule of Signs is what's intended.

Suppose that we have a polynomial p(x) with real coefficients.

The rule says that the number of positive real roots is equal to the number of sign changes (as we move left to right along the polynomial) or is less than that number by an even integer.

Replacing x by -x to get the polynomial p(-x), a similar statement can be made about the number of negative real roots.

For example, the polynomial

\(\displaystyle p(x) = 3x^{5}-2x^{4}+7x^{2}-9 \\ p(-x)=-3x^{5}-2x^{4}+7x^{2}-9,\)

will have 3 or 1 positive and 2 or 0 negative real roots.

Tiggsy Jan 12, 2023