Final answer:
The complete factorization of the polynomial function f(x) = x³ - 4x² + 4x - 16 over the set of complex numbers is (x - 2)(x - (1 + √7i))(x - (1 - √7i)).
Step-by-step explanation:
The student has asked for the complete factorization of the polynomial function f(x) = x³ - 4x² + 4x - 16 over the set of complex numbers. To factor this polynomial, we first look for any common factors or patterns, such as the difference of squares or other recognizable factorization forms. However, this polynomial does not have an immediately obvious factorization pattern, so we will try to find one of the roots to help us in factoring it.
We may apply the rational root theorem or use synthetic division to find a root. If we evaluate the polynomial at x=2, we see that f(2) = 2³ - 4(2)³ + 4(2) - 16 = 0, which indicates that (x - 2) is a factor. Performing long division or synthetic division of f(x) by (x - 2) gives us a quadratic polynomial which can be factored further or solved for roots using the quadratic formula.
Upon division, we get x² - 2x + 8, which does not factor nicely over the integers. However, we can complete the square or directly use the quadratic formula to find the roots of this quadratic, which are complex. The solutions are x = 1 ± √(1² - 8) = 1 ± √(-7), which gives us the complex roots 1 + √7i and 1 - √7i. Therefore, the complete factorization over the complex numbers is:
f(x) = (x - 2)(x - (1 + √7i))(x - (1 - √7i)).