Generalizing the Difference of Squares Formula: Why Don’t We Teach This?

In high school, inevitably students are taught the formula for expressing a difference of squares in terms of linear factors, specifically that

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

This can be a useful factorization method, and while students do not derive it, they can easily verify it by expansion of the right hand side.

Some students will also learn the formula for the sum and difference of squares:

\displaystyle x^3 + y^3 = (x+y)(x^2-xy+y^2) \; \; \& \; \; x^3 - y^3 = (x-y)(x^2+xy+y^2)

Again, verification is simple, even if one does not derive the formula. Moreover, while the formula is longer, memorization is made easy by the mnemonic SOAP, which dictates the signs (“Same Opposite Always Positive”).

Likely even fewer students, presumably in pre-calculus where complex number arithmetic is discussed in more detail than in previous coursework, may even know the sum of squares formula:

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

Once more, verification is a trivial exercise.

But, why teach all these different formulas when they can be combined? Of course, most students do not ask this question, because they do not know that they can be, but in fact, they can.

Theorem. Let x,y \in \mathbf{R} and let n be some natural number, then

x^n-y^n = (x-y)( x^{n-1}+x^{n-2} y + \cdots + xy^{n-2} + y^{n-1})

The best part of this is that its proof is elementary and can be one of the first examples used in the introduction of mathematical induction, which really ought to be taught to all high schoolers at some point.

Proof.

The base case clearly works. So, suppose it works for n =k, then we wish to show it necessarily works for k+1 and hence the formula is valid for all k \in \mathbb{N}.

Let us restate our induction hypothesis as follows:

\displaystyle \sum_{i=0}^{k-1} x^i y^{(k-1)-i} = \frac{x^k-y^k}{x-y}

Now, we proceed as follows with the k+1 case. (The image is via Codecogs’ editor, as I am still getting used to WordPress’ \LaTeX editor.)

CodeCogsEqn (1)

which proves the theorem.

\blacksquare

The only tricky part here is the alternative form of the induction hypothesis. We do this so the induction problem seems more familiar and additive in nature. After that observation, we proceed just as with most classic cases of induction in elementary algebra and number theory.

And, what do we get from this? Well, we get all of the forms of difference of powers we could possible have. Personally, I prefer this to memorizing multiple formulas.

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