In this series of posts, I’ve explored ways that students can discover the formula for the difference of two squares and the difference of two cubes:
If students have understood the origins of these two formulas, then it’s not much of a stretch for students to guess the formula for . A geometric derivation requires four-dimensional visualization which is beyond of what can be reasonably expected of high school students. Still, students can look at the above two formula and guess that is a factor of , and that the second factor would contain and :
From this point forward, it’s a matter of either using long division to find the quotient of or else just guessing (and confirming) the nature of the .
Once students recognize that the answer is
then the factorings of , , etc. become obvious.