Final answer:
The polynomial f(x) of degree 5 with real coefficients given the zeros 1, i, and 2i, must also include the zeros -i and -2i. Without additional information, the fifth zero cannot be determined.
Step-by-step explanation:
We need to find the remaining zeros of a 5th-degree polynomial given the zeros 1, i, and 2i. Since the coefficients are real numbers, the non-real zeros must occur in conjugate pairs. Therefore, if i and 2i are zeros, their conjugates, -i and -2i, are also zeros of the polynomial.
Now, we have four zeros: 1, i, -i, and 2i. Since the polynomial is of degree 5, there should be one more zero. We already have the zero 1, so there is no conjugate pair to consider for 1. Therefore, we count 1 as one real zero, leaving the remaining zero to be discovered.
However, without additional information, such as the polynomial's coefficients or a specific value for the polynomial at a given point, we cannot uniquely determine the remaining fifth zero. If there is more information given, we can apply it to find the last zero.