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 1+5i are​ zeros;

f (1)=100

Answer 1

zeros -3 and 1+5i

x+3 is one factor

x=1+5i

x -1 = 5i

x^2 -2x +1 = -25

x^2 -2x +26 is the other factor

multiply the 2 factors together

f(1) = 100 = a(x+3)(x^2-2x+26) solve for a

a(x^3 +x^2 +20x +78) = 100 when x=1

100a = 100

a=1

f(x) = x^3 +x^2 +20x +78