How many turning points can a polynomial of degree 5 have at most?

A polynomial of degree 5 can have at most 4 turning points. This is because the maximum number of turning points a polynomial can have is one less than its degree. So, in this case, if you take 5 (the degree) and subtract 1, you get 4! This means that as you sketch or analyze the graph of a degree 5 polynomial, you can expect it to zigzag a bit with up to 4 peaks or valleys before settling into its end behavior.

Now, it’s interesting to note that while you can have 4 turning points, achieving the maximum number in practice often depends on the specific coefficients and roots of the polynomial. Some polynomials might dance around having fewer turning points, so the key is to really inspect their nature and behavior!