Final answer:
The Rational Root Theorem is used to find possible rational roots for the polynomial equation x³ + 7x² - 10x - 24 by listing the factors of the constant term and checking each as a potential root.
Step-by-step explanation:
The question asks to use the Rational Root Theorem to find the possible rational roots for the polynomial equation x³ + 7x² - 10x - 24.
The Rational Root Theorem states that if a polynomial equation with integer coefficients has any rational solutions (or roots), those solutions must be of the form ±p/q, where p is a factor of the constant term (in this case, -24) and q is a factor of the leading coefficient (in this case, 1).
To apply the theorem to the given polynomial, we first list the factors of -24, which are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24. Since the leading coefficient is 1, this simplifies the process, as we only need to consider each of these factors (positive and negative) as possible roots.
Therefore, the possible rational roots for this polynomial could be ±1, ±2, ±3, ±4, ±6, ±8, ±12, or ±24. The next step is to test these possible roots using synthetic division or direct substitution to see if any of them yield a zero remainder, confirming that they are indeed roots of the equation.