Final answer:
The cubic polynomial function with zeros at -5, -2, and 2, and a value of -24 at x = -1 is f(x) = 4x^3 + 8x^2 - 36x - 40 in expanded form.
Step-by-step explanation:
The student is asking to write a cubic polynomial function f(x) in expanded form, with given zeros at -5, -2, and 2, and a value at x = -1 as -24. To find the cubic polynomial, we start with the factored form which is (x - x1)(x - x2)(x - x3), where x1, x2, and x3 are the zeros of the polynomial. The expanded form with these zeros is (x + 5)(x + 2)(x - 2). However, this polynomial's value at x = -1 is not necessarily -24, so we must find a constant k such that when f(-1) is calculated, it equals -24.
First we substitute the zeros into the polynomial:
f(x) = k(x + 5)(x + 2)(x - 2)
Then we use the given point to find k:
f(-1) = k(-1 + 5)(-1 + 2)(-1 - 2) = -24
Solving for k, we find:
k(-6) = -24
k = 4
Therefore, the polynomial function is:
f(x) = 4(x + 5)(x + 2)(x - 2)
Which in expanded form is:
f(x) = 4x^3 + 8x^2 - 36x - 40