n=3; -3 and 4+4i are zeros; f(2)=100

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.

Answer 1

Since the polynomial has real coefficients, its complex-valued roots come in complex-conjugated pairs. Therefore, the root 4+4i comes together with the complex-conjugated root 4-4i. These two roots give the factor of (x-4-4i)(x-4+4i) = x² -8x+32 in the polynomial. The real-valued root -3 gives the factor of (x+3). Thus, the requested 3rd-degree polynomial is P₃(x) = (x+3)(x² -8x+32) = x³-5x²+8x+96.