The complex number 5 + 2i is a zero of a rational function. The graph of the function has local maxima at (-1, 0), (2, 0), and (8, 0). What is the least possible degree of the function?

 To determine the least possible degree of a rational function with the given zeros and local maxima, we should follow these steps:

1. Consider the Complex Zero: Since complex zeros of polynomials with real coefficients come in conjugate pairs, if 5 + 2i is a zero, then its conjugate, 5 - 2i, must also be a zero.

2. Real Zeros from Local Maxima: Each local maximum on the graph suggests that there could be a real zero at that x-coordinate. However, a local maximum does not necessarily mean a zero is exactly at that point; instead, it might indicate a sign change of the derivative, which implies a possible zero nearby.

3. Counting the Zeros: At this stage, we have two complex zeros (5 + 2i and 5 - 2i) and potentially three real zeros corresponding to the local maxima at (-1, 0), (2, 0), and (8, 0).

4. Degree of the Polynomial Part: Each zero corresponds to a factor in the polynomial. Therefore, having two complex zeros contributes a degree of 2, and each real zero contributes a degree of at least 1.

5. Minimum Degree: Assuming each local maximum corresponds to an actual zero, the polynomial part of the rational function would have at least 5 zeros in total (two complex and three real), which implies a minimum degree of 5.

However, we need to be cautious: the function is a rational function, so if the numerator and denominator share any factor, that factor would not count as a zero since it would be canceled out. But as we have no information on the denominator of the rational function, we can only infer information based on its numerator.

Therefore, the least possible degree of the polynomial part of the rational function, given the information provided, is 5.