F(x)=5x^3+29x^2+19x-5

Question

Use Descartes rule of signs to determine the number of positive and negative real zeros

Answer 1

Descartes' Rule is that the number of positive real zeros in a polynomial function is equal to or less than by an even number the number of changes in the sign of the coefficients. There's only one change of sign. Therefore the number of positive real zeroes≤1.

By observation, x=-1 is a real zero because f(-1)=0.

Divide by this zero:

-1 | 5 29  19  -5

      5  -5 -24 | 5

      5 24   -5 | 0 = 5x²+24x-5=(x+5)(5x-1).

Zeroes: -1, -5, ⅕. (2 negative zeroes and one positive).