Solve the cubic equationx3 − x2 − 14x + 24 = 0.

Sure, let's solve the cubic equation step by step.

The given equation is x³ - x² - 14x + 24 = 0.

Step 1: Look for possible rational roots using the Rational Root Theorem. This theorem states that any rational root, p/q, of the polynomial must have p as a factor of the constant term (24 in this case) and q as a factor of the leading coefficient (1 in this case). So, the possible rational roots of the equation are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.

Step 2: Use synthetic division or direct substitution to test the possible roots. If a number is a root, the remainder will be zero.

By trying these values, we find that x = 1, x = -2, and x = 3 are the roots of the equation.

So, the solutions to the equation x³ - x² - 14x + 24 = 0 are x = 1, x = -2, and x = 3.