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 5 i  are​ zeros;  f(2)=116

Answer 1

If 5i is a zero then so is -5i. (Complex roots come in conjugate pairs.)

The 3rd degree polynomial has the form f(x)=a(x-4)(x²+25).

When x=2, f(x)=116, so a(-2)(29)=-58a. Therefore a=-116/58=-2.

f(x)=-2(x-4)(x²+25)=-2(x³+25x-4x²-100),

f(x)=-2x³+8x²-50x+200.