Challenge Level

(a) Evaluate $(x + x^2 + x^3 + x^4)^2$.

(b) Imagine you have two spinners labelled $1, 2, 3$ and $4$ and spin them together. The score is the sum of the results from the two spinners. Find the theoretical frequency distribution of the scores.

(c) What do you notice about this frequency distribution and the coefficients in the polynomial expansion from (a)?

(d) Notice that the powers in $(x^1 + x^2 + x^3 + x^4)$ correspond to the labels on the spinners. Can you factorize the expression $(x + x^2 + x^3 + x^4)^2$ into two different polynomials which correspond to a re-labelling of the spinners, so each has four non-negative integer labels, giving new pairs of spinners with the same frequency distribution of scores? This re-labelling can be done
in more than one way.

(e) Find other pairs of spinners which can be re-labelled in more than one way to give the same frequency distribution of scores.

What about a $2$-spinner and a $3$-spinner? What about two ordinary dice?