Final answer:
Descartes' Rule of Signs applied to the polynomial indicates that there can be up to 3 positive real roots and up to 4 negative real roots, with the actual numbers possibly being less by an even number.
Step-by-step explanation:
Descartes' Rule of Signs is used to determine the possible number of positive and negative real roots in a polynomial. For the polynomial x⁵+x⁴-13x³-23x²-14x-24, we can analyze the sign changes to apply this rule.
When evaluating the polynomial with a + sign for x, changes in the signs of the terms from positive to negative or vice versa are counted. There are 3 sign changes (x⁴ to -13x³, -13x³ to -23x², and -14x to -24), indicating up to 3 positive real roots.
For the - sign for x, we substitute x with -x in the polynomial and count the sign changes. The transformed polynomial is (-x)⁵ + (-x)⁴ - 13(-x)³ - 23(-x)² - 14(-x) - 24, which simplifies to -x⁵ + x⁴ + 13x³ - 23x² + 14x - 24. There are 4 sign changes, indicating up to 4 negative real roots.
It's important to remember that Descartes' Rule gives the maximum number of positive or negative real roots, and that the actual number may be less by an even number. Therefore, the polynomial could have 3 or 1 positive roots and 4 or 2 negative roots.