Final answer:
To solve the cubic equation x³ - x² - 16x + 16, we can use the Rational Root Theorem to find the possible rational roots and synthetic division to test them. The solutions to the equation are x = 2, x = (3 + √41) / 2, and x = (3 - √41) / 2.
Step-by-step explanation:
To solve the cubic equation x³ - x² - 16x + 16, we can use the Rational Root Theorem to find the possible rational roots. The possible rational roots are factors of 16, which are ±1, ±2, ±4, ±8, and ±16. By checking these possible roots using synthetic division, we find that the root x = 2 is a solution. We can then use polynomial division or synthetic division to divide the given cubic equation by (x - 2) to obtain a quadratic equation. Solving this quadratic equation will give us the other two roots.
The quadratic equation obtained is x² - 3x - 8 = 0. To solve this equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Substituting the values a = 1, b = -3, and c = -8 into the formula, we get x = (-(-3) ± √((-3)² - 4(1)(-8))) / (2(1)). Simplifying further, we have x = (3 ± √(9 + 32)) / 2, which gives us x = (3 ± √41) / 2. Therefore, the solutions to the cubic equation x³ - x² - 16x + 16 are x = 2, x = (3 + √41) / 2, and x = (3 - √41) / 2.