Question

If a graph touches the x axis at -3 and crosses at 6 and is a polynomial function of (x-6)^3 (x+3)^2 , then what is the maximum number of turning points on the graph?

Answer 1

A turning point is a point where the graph changes from increasing to decreasing. The maximum number of turning points that a polynomial can have is always 1 less than the degree of the polynomial.

If a polynomial is given by the expression:

`P(x)=(x-6)^3(x+3)^2`

Then, the degree of the polynomial is 5. Then, the maximum number of turning points that the polynomial could have is 4.

However, the graph of the polynomial actually looks like:

As we can see, the actual number of turning points is 2.

Therefore, the answer is: the maximum number of turning points that the graph could have is 4.