(1) Take any monic quadratic polynomial, i.e. an expression of the form x² + bx + c.

Pick any three consecutive integers, e.g. 4, 5, 6 or -9, -8, -7.

Substitute your largest and smallest integers into your quadratic and add the results. Call this A.

Then substitute your middle integer into your polynomial and double the result. Call this B.

Finally, evaluate A - B. What do you notice? Can you prove that it always works?

(2) Can you adapt the trick to work with a monic cubic polynomial, i.e. x³ + bx² + cx + d?

Can you generalise to a polynomial of any degree?