Doug C. answered • 09/05/19
Math Tutor with Reputation to make difficult concepts understandable
Complex roots come in pairs called conjugates. That means if 4i is a root, then so is -4i.
Therefore (x-3), (x-4i), and (x+4i) are factors of the polynomial function. There also could be a constant factor (which the fact that f(1) = 102 will help to determine.
Multiplying (x-4i)(x+4i) results in a difference of squares: x2 - 16i2 = x2 + 16. Multiply that result by (x-3) to get:
x3 - 3x2 + 16x -48.
So we have f(x) = A(x3 - 3x2 + 16x -48).
When x = 1, f(1) = 102, so 102 = A(-34), and A = -3.
Finally we have f(x) = -3x3 + 9x2 -48x + 144, after distributing the A=-3 times each term.
If you know synthetic division you can confirm that (x-3) is a factor by using that process and showing a remainder of 0. The coefficients that remain show that (-3x2 -48) as the quotient. And solving that for x when it is set to 0 results in +/- 4i.
