Final answer:
To factor the polynomial function completely, we can use the complex conjugate roots theorem. We start by setting up two equations, one for the given zero 4-2i and another for its conjugate 4+2i. We then multiply the factors to obtain a quadratic equation. Factoring the quadratic equation gives us the complete factorization of the polynomial function.
Step-by-step explanation:
The given polynomial function is f(x) = x⁴-9x³-8x²-140x-400. The zero 4-2i tells us that (x - (4-2i)) = 0. Based on the complex conjugate roots theorem, we know that if 4-2i is a zero, then its conjugate 4+2i is also a zero. So, (x - (4+2i)) = 0. Expanding the binomials and simplifying, we get (x - 4 + 2i)(x - 4 - 2i) = 0. Multiplying these two factors, we obtain (x² - 8x + 16 + 4) = 0. Simplifying further, we get x² - 8x + 20 = 0. Now, we need to factor the remaining quadratic equation. Factoring, we find (x - 2)(x - 10) = 0. Therefore, the completely factored polynomial function is f(x) = (x - 4 + 2i)(x - 4 - 2i)(x - 2)(x - 10).