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.

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*Posted by John Quintanilla on June 18, 2016*

https://meangreenmath.com/2016/06/18/difference-of-two-powers-part-5/

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