Question

section 3.5 - factoring polynomials over the complex numbers

factor the polynomial completely over the complex numbers.

(f(x) = x^4 - 16)

Explanation:

Apply difference of squares

$$ f(x) = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) $$

Factor real quadratic term

$$ x^2 - 4 = (x - 2)(x + 2) $$

Factor complex quadratic term

$$ x^2 + 4 = x^2 - (2i)^2 = (x - 2i)(x + 2i) $$

Answer:

\(f(x) = (x - 2)(x + 2)(x - 2i)(x + 2i)\)