Final answer:
The polynomial function in factored form, given the roots and the y-intercept, is f(x) = -(x + 2)(x - 2)(x + 3). The standard form after expanding the factors is f(x) = -x³ - 3x² + 4x + 12.
Step-by-step explanation:
Given that f(x) is a third-degree function that passes through the point (0, 12) and has roots at -2, 2, and -3, we can write the polynomial in its factored form as follows:
f(x) = a(x + 2)(x - 2)(x + 3)
To find the value of a, we use the fact that f(0) = 12.
Substitute x with 0:
12 = a(0 + 2)(0 - 2)(0 + 3)
12 = -12a
a = -1
The factored form of the polynomial is:
f(x) = -(x + 2)(x - 2)(x + 3)
To convert to standard form, expand the factors:
f(x) = -[(x² - 4)(x + 3)]
f(x) = -(x²x + 3x² - 4x - 12)
f(x) = -x³ - 3x² + 4x + 12
The standard form of the polynomial is:
f(x) = -x³ - 3x² + 4x + 12