Find the complex zeros of the function

Question

f(x)=x^4-4x^3+8x^2-16x+16

Answer 1

By the Rational Root Theorem, the possible rational roots are:

±1, ±2, ±4. ±8, ±16

Using synthetic substitution 2 is a root leaving the suppressed polynomial

x³ - 2x² + 4x - 8

This can be factored into

x²(x - 2) + 4(x - 2)

(x² + 4)(x - 2)

 

2 is the second root. The third and fourth roots are 2i and -2i.