College Algebra

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

7+2i

are​ zeros;

f (−1)=204

Answer 1

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.

1. n = 3 ; -4 and 7 + 2i are zeros; f(-1) = 204

Since f(x) has real coefficients 7+2i is a root, so is 7-2i

So, -4, 7+2i, and 7-2i are roots

Since f(x) has degree 3, there can't be more than 3 distinct roots.

f(x) = a(x+4)(x-7-2i)(x-7+2i)

f(x) = a(x+4)(x²-14x+53)

f(-1) = 204, so

204 = a(-1+4)(-1²-14(-1)+53)

204 = a(+3)(1+14+53)

204/204 = a

1 = a

f(x) = 1(x+4)(x²-14x+53)

You can use your graphing calculator (or desmos) to verify the real zero at x = 2

I hope this helps you!