Students will learn how to add and subtract polynomials. Students will also lead to the following conclusion: Let \(u\) be the degree of polynomial \(P,\) and let \(v\) be the degree of polynomial \(Q.\) Then the degree of \(P + Q,\) \(P - Q,\) or \(Q - P\) cannot be greater than \(\max(u, v).\) You can lead students to this conclusion by asking them to add and subtract various polynomials, and asking students for the degree instead of the sum or difference. They should also see the degree isn't necessarily equal to \(\max(u, v).\) For example, suppose \(P = Q = x^2.\) Then \(P - Q = 0,\) which has degree \(0.\) But \(\max(u, v) = 2.\) Thus, we have found a case where the degree of \(P - Q\) is not equal to \(\max(u, v).\) The same is reasoning works for \(P + Q,\) by setting \(Q = -P.\)

Conclude by giving your students these challenges:

- It Was 2010! by NRICH
- Seating Arrangements by NRICH
- 2009 AMC 8, Problem 18
- Balancing Equations by Pierce School: Problem / Solution

- I can reverse 12 by adding 9 to it: 12 + 9 = 21.
- I can reverse 15 by adding 36 to it: 15 + 36 = 51.
- What must I add to 141414 to reverse it?

Here's the solution.

Note: I rewrote the problem because I thought its presentation was hokey.