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 4i are zeros; f(1)=34

Step 1 ：We are given that the polynomial is of degree 3 and has roots 2, 4i, and -4i. The general form of a polynomial is given by \(f(x) = a(x - r1)(x - r2)(x - r3)\), where \(r1, r2, r3\) are the roots of the polynomial and \(a\) is a constant.

Step 2 ：Substituting the roots into the equation, we get \(f(x) = a(x - 2)(x - 4i)(x + 4i)\).

Step 3 ：We also know that \(f(1) = 34\). Substituting \(x = 1\) into the equation, we can solve for \(a\).

Step 4 ：Doing so, we find that \(a = -2\).

Step 5 ：Substituting \(a\) back into the equation, we get the final polynomial function: \(f(x) = -2(x - 2)(x - 4i)(x + 4i)\).

Step 6 ：Thus, the nth-degree polynomial function with real coefficients satisfying the given conditions is \(\boxed{-2(x - 2)(x - 4i)(x + 4i)}\).