A turning point is when the gradient changes sign, namely a local maximum or minimum. An nth degree polynomial can have up to n-1 turning points.
For example, a straight line (first degree polynomial) does not have a turning point, since 1-1=0. A parabola is a second degree polynomial, therefore it has 2-1=1 turning point.
If a polynomial has 5 turning points, it has a minimum degree of 5+1=6.
When two polynomials are multiplied together, the resulting degree of the product is the sum of the degrees of the original polynomials.
