Difference of Two Powers (Part 5)

Algebra II 1 Minute

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:

x^2 - y^2 = (x-y) (x+y)

x^3 - y^3 = (x-y)(x^2 + xy + y^2).

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 x^4 -y^4. 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 x-y is a factor of x^4-y^4, and that the second factor would contain x^3 and y^3:

x^4 - y^4 = (x-y)(x^3 + \hbox{~~~something~~~} + y^3).

From this point forward, it’s a matter of either using long division to find the quotient of x^4-y^4 or else just guessing (and confirming) the nature of the \hbox{something}.

Once students recognize that the answer is

x^4 - y^4 = (x-y)(x^3 + x^2 y + x y^2 + y^3),

then the factorings of x^5 - y^5, x^6 - y^6, etc. become obvious.

Published by John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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One thought on “Difference of Two Powers (Part 5)”

  1. Pingback: Difference of Two Powers (Index) | Mean Green Math

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