Answer:
f(x) = x⁴-11x³+37x²-29x-78
Explanation:
The zeros -1 and 6 are real zeros, so we'll use them directly.
The zero 3-2i is complex, and ALL complex zeros come in pairs. So we use the complex conjugate of 3-2i for the other zero: 3+2i.
Now, set up your factors. There are four zeros, so four factors:
(x- -1)(x-6)(x-(3-2i))(x-(3+2i)) = (x+1)(x-6)(x- 3-2i)(x- 3+2i)
Next, the instructions say "with real coefficients" so we must multiply everything out and combine like terms. I recommend starting with the two real factors: (x+1)(x-6) = x²-5x-6.
then multiply the two complex factors:
(x- 3-2i)(x- 3+2i) = x²-3x+2ix-3x+9-6i-2ix+6i-4i² [remember that i²=-1!!]
= x²-6x+13
and finally multiply the two trinomials that you get; you'll have 9 terms to combine.