Final answer:
Another zero of R(x) can be 4-5i, -3+3i, +i based on the given zeros and the Complex Conjugate Root Theorem. The maximum number of real zeros R(x) can have is 3. The maximum number of nonreal zeros it can have is 6.
Step-by-step explanation:
Finding Zeros of a Polynomial
Part (a): Another zero of R(x)
Given that R(x) is a polynomial with real coefficients and has a zero of 4+5i, by the Complex Conjugate Root Theorem, its conjugate 4-5i must also be a zero. Similarly, -3-3i implies another zero, -3+3i, and -i implies its conjugate +i as zeros of R(x).
Part (b): Maximum number of real zeros
The degree of the polynomial gives the maximum number of zeros it can have. A degree 9 polynomial can have at most 9 zeros. Since complex zeros occur in conjugate pairs, and we have identified two pairs (4+5i & 4-5i, -3-3i & -3+3i) and one pair (-i & +i), the maximum number of real zeros is 9 - (2 x 2+2 x 1) = 9 - 6 = 3.
Part (c): Maximum number of nonreal zeros
The polynomial can have at most 9 zeros, with each nonreal zero having a conjugate pair. Therefore, the maximum number of nonreal zeros is 6, as there can be three pairs of nonreal zeros (the ones already identified).