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 4i are zeros;
f(-1)=85

F(x) = ________

Answer 1

Givem that at n = 3: 4 and 4i are zero;

Then;

$(x - 4)(x - 4i)(x + 4i) \: (x - 4)( {x}^{2} - ( {4i)}^{2} \\ (x - 4)(x - 4i)(x + 4i) \: (x - 4)( {x}^{2} + 16) \\ (x - 4)(x - 4i)(x + 4i) \: {x}^{3} + 16x - {4x}^{2} - 64 \\ (x - 4)(x - 4i)(x + 4i) \: {x}^{3} - {4x}^{2} + 16x - 64$

Hence the function is:

$f = {x}^{3} - {4x}^{2} + 16x - 64$