Find The Cubic Polynomial With Zeros 3 And 3 4i And 3 4i And Having F 1 64

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+4i are zeroes f(1)=64

Solution

Since 3+4i is a root, 3-4i must also be a root.   f(x) = a(x-3)[x-(3+4i)][x-(3-4i)]         = a(x-3)[(x-3)-4i][(x-3)+4i]         = a(x-3)[(x-3)2-(4i)2]         = a(x-3)(x2-6x+25)   Since f(1) = 64, -40a = 64.  So, a = -8/5   f(x) = (-8/5)(x-3)(x2-6x+25)