Final answer:
To factor the polynomial P(x) = x⁴ - 2x³ - 8x + 16 and express it in factored form, we can use the factor theorem and synthetic division. The factored form of P(x) is (x + 2)(x² - 4x + 8).
Step-by-step explanation:
To factor the polynomial P(x) = x⁴ - 2x³ - 8x + 16 and express it in factored form, we can use the factor theorem and synthetic division. The first step is to find the possible rational roots of the polynomial. By using the rational root theorem, we can determine that the possible rational roots are ±1, ±2, ±4, ±8, ±16. By checking these values, we find that x = -2 is a root of the polynomial.
Using synthetic division with -2 as the root, we divide the polynomial by (x + 2) to obtain the polynomial (x + 2)(x² - 4x + 8).
The factored form of P(x) is P(x) = (x + 2)(x² - 4x + 8).