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

4 and 2i are zeros

f(-1)=50

Answer 1

To have real coefficients, if a complex number is a root, so is its complex conjugate. Thus, if 2i is a root, so is -2i.

Thus, our polynomial is a(x-4)(x^2 + 4)

Then, if f(-1) = 50,

a(-1-4)((-1)^2 + 4) = 50

-25a = 50

a = -2

f(x) = -2(x-4)(x^2 + 4) = -2x^3 + 8x^2 - 8x +32