Final answer:
The number of possible positive real roots of the equation x^5 - 4x^4 + 5x^3 + 9x^2 - 8 is either 2 or 0, and the number of possible negative real roots is either 3 or 1, as predicted by Descartes' Rule of Signs.
Step-by-step explanation:
To determine the number of possible positive and negative real roots of the polynomial equation x^5 - 4x^4 + 5x^3 + 9x^2 - 8, we can use Descartes' Rule of Signs. First, let's look at the number of sign changes in the original equation to predict the number of positive real roots. Every time the coefficient of a term in the polynomial goes from positive to negative or vice versa, this is counted as a sign change.
In x^5 - 4x^4 + 5x^3 + 9x^2 - 8, there are two changes in sign (from x^5 to -4x^4 and from 9x^2 to -8), therefore, there can be 2 or 0 positive real roots (since the number of positive real roots is either the number of sign changes or that number minus an even number).
For negative real roots, we substitute x with -x and check the sign changes again. Our equation becomes ((-x)^5 - 4(-x)^4 + 5(-x)^3 + 9(-x)^2 - 8) which simplifies to -x^5 - 4x^4 - 5x^3 + 9x^2 - 8. In this form, there are three sign changes, which means there can be 3 or 1 negative real roots.