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 6+4i are zeros;
f(2)=128

Answer 1

f(x) = x³ -10x² +28x +104

Explanation:

You want a cubic polynomial f(x) with real coefficients, and its graph, such that one root is -2, another is 6+4i, and f(2) = 128.

Conjugate roots

A polynomial with real coefficients will have complex (or radical) roots that are conjugate pairs. That means the root 6+4i will be paired with a root 6-4i. The factored form of the required polynomial will be ...

f(x) = a(x -(-2))(x -(6 +4i))(x -(6 -4i))

f(x) = a(x +2)(x² -12x +52)

Scale factor

We can use the given value of f(2) to find the scale factor, a.

f(2) = 128 = a(2 +2)(2² -12·2 +52) = a(4)(32) = 128a

a = 1 . . . . . divide by 128

The required polynomial is ...

f(x) = (x +2)(x² -12x +52)

f(x) = x³ -10x² +28x +104