find an nth degree

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=4
-1,5,and 4+3i are zeros;
f(1)=-144

Answer 1

There are 2 facts you need to know to solve this problem.

1> Since 4 + 3i is a root and the equation has real coefficients, 4 - 3i must also be a root.

2> The only parameter you have to make f(1) = -144 is a multiplier for the entire polynomial.

 

The equation you want is A(x+1)(x-5)(x² -8x + 25) but f(1) = 144A and therefore A=1.

 

Since the problem did not require it, I did not multiply to get all of the coefficients of the 4th degree polynomial.