Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.

Question

Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing​ utility, use it to graph the function and verify the real zeros and the given function value.

n=3;

2 and 2 i are zeros;

f(-1)=15

f(x)=

Answer 1

Use the factored form. Since the degree n=3, you need three factors:

f(x) = a·(x-p)(x-q)(x-r)

where a is a constant and p, q, and r are the zeros. You are given two of the zeros, 2 and 2i. With real coefficients, the Conjugate Root Theorem tells us that -2i must also be a zero. So we have our three zeros:

f(x) = a·(x-2)(x-2i)(x+2i)

To find the value of the constant a, plug in the given point (x,y) = (-1,15) and solve for a.