Question

Using the given zero, 2 - 4i, find one other zero of f(x) = x4 - 4x3 + 21x2 - 4x + 20

Answer 1

The given function is
f(x) = x⁴ - 4x³ + 21x² - 4x + 20

A zero is 2 - 4i. Because it is complex, it should have a conjugate of 2 + 4i.
The product of the two complex conjugates yield
(2 - 4)*(2 + 4i) = 2² - 4²i² = 4 + 16 = 20

The two complex factors are [x-(2-4i] and [x-((2+4i)].
Therefore a real factor is
x² - x(2+4i+2-4i) + 20 = x² - 4x + 20

Perform long division.
                                     x²          + 1
                ------------------------------------
x²-4x+20 | x⁴ - 4x³ + 21x² - 4x + 20
                  x⁴ - 4x³ + 20x²
                 ----------------------------------
                                      x² - 4x + 20
                                      x² - 4x + 20

Therefore another real factor is (x² + 1).
The complex zeros of x² + 1 are i and -i.

Answer: One other zero is either i or -i.