Find a polynomial of degree 3 with real coefficients and zeros of - 3, - 1, and 4, for which f (-2) = 12.

2 (x + 3) (x + 1) (x - 4)

or 2x^3 - 26x - 24.

Step-by-step explanation:

We can write the polynomial if factor form:

P (x) = a (x + 3) (x + 1) (x - 4) where a is some constant.

Now, since f (-2) = 12 we can write:

12 = a (-2 + 3) (-2 + 1) (-2-4)

12 = 6a

a = 2.

So the polynomial is

2 (x + 3) (x + 1) (x - 4).

Expanded that is

2 (x + 3) (x^2 - 3x - 4)

= 2 (x^3 - 3x^2 - 4x + 3x^2 - 9x - 12)

= 2x^3 - 26x - 24.