Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.

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;

-3 and 3+5i are zeros;

f (1) = 116

f(x)=

Answer 1

The 3 roots of f(x) are 3, 3+5i and 3-5i. By the fundamental theorem of algebra, for some constant k,

f(x)= k (x+3)(x-(3+5i))(x-(3-5i)) = k (x+3)(x^2-6x+34)= k (x^3-3x^2+16x+102). We calculate that

f(1)=k(1-3+16+102)=116k and since it is given that f(1)=116, then k=1 and

f(x) = x^3 - 3x^2 + 16x + 102