How do I solve this problem?

Question

my precalculus homework has this problem: “
x⁵+x⁴-3x³-11x²-28x-60=0 hint:2i is a solution” Im not sure how to solve this

Answer 1

A degree 5 polynomial has 5 roots.  If 2i is a root, so is -2i, so two of the factors are (x+2i)(x-2i) = x²+4.  We have two roots, we need to find the other three

 

1. Divide x⁵ + x⁴ - 3x³ - 11x² - 28x - 60 by (x²+4).  The quotient will be x³ + x² - 7x - 15.
2. Apply the Rational Root Theorem to see if there is a rational root to x³ + x² - 7x - 15.  There is, x = 3.  Divide x³ + x² - 7x - 15 by (x-3).  The quotient is x² + 4x + 5.
3. Use the Quadratic Formula to find the roots of x² + 4x + 5.  The answer is -2 ± i.
4. The roots are ±2i, 3, -2 ± i