he student asked to determine which linear expression is a factor of the polynomial function f(x) = x³ + 2x² - x - 2. To do this, we can use the Factor Theorem, which states that if f(c) = 0, then x - c is a factor of the polynomial. By evaluating the function at the provided options:
f(x) = (x - 1) would give us f(1) = 1 + 2 - 1 - 2 = 0, which is not an option provided.
f(x) = (x - 2) would give us f(2) = 2³ + 2(2²) - 2 - 2 = 8 + 8 - 2 - 2 = 12, so x - 2 is not a factor.
f(x) = (x + 2) would give us f(-2) = (-2)³ + 2(-2)² + 2 - 2 = -8 + 8 - 2 = -2, so x + 2 is not a factor.
f(x) = (x + 3) would give us f(-3) = (-3)³ + 2(-3)² + 3 - 2 = -27 + 18 + 3 - 2 = -8, so x + 3 is not a factor.
f(x) = (x - 3) would give us f(3) = 3³ + 2(3²) - 3 - 2 = 27 + 18 - 3 - 2 = 40, so x - 3 is not a factor.
