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?
What are the possible numbers of negative 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 variations in the sequence of coefficients or is less than that by an even number.
In the given polynomial function p(x) = -x^3 + 5x^2 + 7x + 8, there are two sign variations in the sequence of coefficients (-, +, +, +), which means that there are either 2 positive real roots or 0 positive real roots.
The rule does not give any information about the number of negative real roots directly, but we can apply it to the polynomial f(-x) = -(-x)^3 + 5(-x)^2 + 7(-x) + 8. Simplifying this expression, we get f(-x) = -x^3 + 5x^2 - 7x + 8. Now, we can apply Descartes' Rule of Signs to this polynomial.
In the polynomial f(-x) = -x^3 + 5x^2 - 7x + 8, there are two sign variations in the sequence of coefficients (-, +, -, +), which means that there are either 2 or 0 negative real roots.
Therefore, the possible numbers of positive real roots of the polynomial p(x) = -x^3 + 5x^2 + 7x + 8 are 2 or 0, and the possible numbers of negative real roots are also 2 or 0.