Given that -4i is a zero, factor the following polynomial function completely. Use the conjugate roots theorem, if applicable. F(x)= x^4+x^3+10x^2+16x-96

Question

Given that -4i is a zero, factor the following polynomial function completely. Use the conjugate roots theorem, if applicable. F(x)= x^4+x^3+10x^2+16x-96

Answer 1

Since -4i is a zero, +4i must be as well, by the conjugate roots theorem. Therefore, (x + 4i)(x - 4i)= x² + 16 must be a factor of x⁴ + x³ +10x² + 16x - 96. By long division, x⁴ + x³ +10x² + 16x - 96 = (x² + 16)(x² + x - 6), and the latter factor can itself be factored as (x + 3)(x - 2).

The complete factoring over C is (x + 4i)(x - 4i)(x + 3)(x - 2).

Answer 2

4i is also a zero

(x + 4i)(x - 4i) is a factor of F(x)

Expand and use long division.