Final Answer:
The values of p, q, and r are p = -8, q = -19, and r = -21. This makes f(x) factorable as:
f(x) = (x + 2)(x - 1)(x - 3)
Step-by-step explanation:
Remainder Theorem: When f(x) is divided by (x + 2), the remainder is -48, which implies f(-2) = -48. Similarly, f(1) = 0 and f(3) = 2.
Substitute values into polynomial: Plug these values into the expression for f(x):
f(-2) = (-2)^3 - 8(-2)^2 + q(-2) + r = -48 (solving for q, we get q = -19)
f(1) = 1 - p + q + r = 0 (solving for r, we get r = -21)
f(3) = 27 + 9p + 3q + r = 2 (no new information gained)
Factor by grouping: Since the equation factors perfectly with roots -2, 1, and 3, we can use grouping to find the factors:
f(x) = (x^2 - px + q)(x - r)
Substitute the values of q and r: f(x) = (x^2 - 8x - 19)(x + 21)
Finally, factor the quadratic: f(x) = (x + 2)(x - 1)(x - 3)
Therefore, f(x) can be fully factored as (x + 2)(x - 1)(x - 3) with p = -8, q = -19, and r = -21.