Answer:
f(x) = x^4 - 2x^3 + 21x^2 - 50x + 100.
Step-by-step explanation:
1. For the real zeros 4 and -2, we can write the linear factors as (x - 4) and (x + 2) respectively.
2. For the complex zeros 5i and -5i, we need to create quadratic factors. The conjugate pair 5i and -5i corresponds to the quadratic factor (x^2 + 25).
3. Multiplying all these factors together, we get the polynomial function:
f(x) = (x - 4)(x + 2)(x^2 + 25)
4. Expanding this polynomial, we get:
f(x) = (x^2 - 2x - 4)(x^2 + 25)
f(x) = x^4 + 25x^2 - 2x^3 - 50x - 4x^2 + 100
f(x) = x^4 - 2x^3 + 21x^2 - 50x + 100
Therefore, the polynomial function with real coefficients given the zeros 4, -2, and 5i is f(x) = x^4 - 2x^3 + 21x^2 - 50x + 100.
