# Difference of Two Powers (Part 5)

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.

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