What does Descartes' Rule of Signs say about the number of positive real roots and negative real roots for the polynomial function p(x)=-x^3+5x^2+7x+8
What are the possible numbers of positive real roots?
Descartes' Rule of Signs states that the number of positive real roots of a polynomial function is equal to the number of sign changes in the coefficients or is less than this by an even number.
For the polynomial function p(x) = -x^3 + 5x^2 + 7x + 8, there are two sign changes in the coefficients. Therefore, the possible numbers of positive real roots are 2, 0, or any even number less than these.
According to Descartes' Rule of Signs, the number of negative real roots of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial function p(-x) or is less than this by an even number.
For the polynomial function p(-x) = -(-x)^3 + 5(-x)^2 + 7(-x) + 8, the sign changes in the coefficients are 3, 1, and 0. Therefore, the possible numbers of negative real roots are 3, 1, or any odd number less than these.