A polynomial function p(x) has a relative maximum at (3, -4) a relative minimum at (-1, -6), a relative maximum at (-5, 2) and no other critical points. How many zeros does p(x) have?

To determine the number of zeros a polynomial function has, we need to consider its behavior around the x-axis. Since the given function has a relative maximum, a relative minimum, and no other critical points, we know that the polynomial is changing from increasing to decreasing and then increasing again.

Let's summarize the given information:

Relative maximum: (3, -4)
Relative minimum: (-1, -6)
Relative maximum: (-5, 2)

From this information, we can infer that the polynomial function crosses the x-axis at least once between each pair of consecutive relative extrema.

Therefore, the function must have a zero between the relative maximum at (3, -4) and the relative minimum at (-1, -6), and another zero between the relative minimum at (-1, -6) and the relative maximum at (-5, 2). This gives us a total of at least 2 zeros.

However, since the problem states that there are no other critical points, we know that there are no additional relative extrema between these given points. Therefore, there are exactly 2 zeros for this polynomial function.

In conclusion, the polynomial function p(x) has 2 zeros.