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=34 and 4i are zeros;f(-1)=85F(x) = ________

Answer 1

Given that at n=3: 4 and 4i are zeros:

Then:

`\begin{gathered} (x-4)(x-4i)(x+4i)\Rightarrow(x-4)(x^2-(4i)^2) \\ (x-4)(x-4i)(x+4i)\Rightarrow(x-4)(x^2+16) \\ (x-4)(x-4i)(x+4i)\Rightarrow x^3+16x-4x^2-64 \\ (x-4)(x-4i)(x+4i)\Rightarrow x^3-4x^2+16x-64 \end{gathered}`

Hence the function is:

`F=x^3-4x^2+16x-64`