A polynomial is to be constructed that will touch the x axis at most 8 times. What is the minimum degree of the polynomial?

The maximum number of roots to a polynomial of order n is n roots. Take the example of a quadratic (order 2) which can intersect the x-axis a maximum of 2 times, and similarly a cubic (order 3) 3 times maximum.

Hence for 8 intersections, minimum order = 8