Given the polynomial x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1 what are the characteristics of its roots.

The polynomial given is f(x) = x^20 - x^19 + 3x^18 - 2x^17 - x^16 + 3x^15 + x^14 + 2x^10 - x^9 + x^6 - 8x^4 + x^3 - 1

f(x) = +x^20 - x^19 + 3x^18 - 2x^17 - x^16 + 3x^15 + x^14 + 2x^10 - x^9 + x^6 - 8x^4 + x^3 - 1

The sign changes are: +-, -+, +-, --, -+, ++, ++, +-, -+, +-, -+, +-

Using Descartes' rule of signs as there are 9 sign changes , there can be a maximum of 9 positive real roots.

f(-x) = +x^20 + x^19 + 3x^18 + 2x^17 + x^16 - 3x^15 + x^14 + 2x^10 - x^9 + x+6 - 8x^4 - x^3 - 1

Here, the sign changes are: ++, ++, ++, ++, +-, -+, ++, +-, -+, +-, --, --

As there are 5 sign changes, there can be a maximum of 5 negative real roots.

The given polynomial can have a maximum 9 positive real roots and 5 negative real roots.