I confess: I think polynomial long division is kind of a waste of time. It’s a tedious process that doesn’t really involve much mathematical understanding. And when you use synthetic division, there’s even less understanding involved. So I say just skip it.
Unfortunately for me, my school’s current FST (2nd half of Alg 2) curriculum includes polynomial long division. The reason is so that we can factor and solve equations like y = 9x^3 – 31x -10 …but I’m not entirely convinced that that’s very useful either. Math class needs to move behind problems that wolfram alpha can solve for us in 3 seconds.
So back to teaching polynomial long division. It actually went well. I really emphasized CCSS Standard for Mathematical Practice #3: Make sense of problems and persevere in solving them. I told my FST kids that there are problems in math (and in life) that are long and challenging and that require stamina and perseverance. For example, these long division problems will test your mathematical stamina, but stick with it and don’t give up.
So many of them took that as a challenge. They wanted to prove that they could stick with the problem all the way through. It was lovely. So maybe there is something to be said about polynomial long division after all.
It was also great when I told them to use zero placeholders for “missing” terms (like 0x^2 in my example above) because right after I said that I forgot to use a placeholder in my example, so then it became completely clear why placeholders are useful when terms weren’t lining up. Yay for making mistakes.