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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Quotients & Homomorphisms for Beginners: Part 1, Motivation

In a typical first course in “abstract” or “modern” algebra (known simply as “algebra” to mathematicians), we introduce students to quotient groups and homomorphisms. In fact, the quotient group is the first group one comes across that is somewhat complicated; most other groups discussed up to that point will be abelian, cyclic, et cetera, and most importantly, students will be at least somewhat familiar and comfortable with the groups beforehand. Therefore, the quotient group represents a unique challenge, a first barrier for students with respect to group theory.

I believe many educators and students would agree with my stance that the quotient group is rarely well motivated, and even more rarely well understood by students, at least while they are in the course (for many, it will ‘click’ later on). This is unfortunate for many reasons. First, the aforementioned challenge is a massive intellectual achievement for those who do grasp the structure. Second, the quotient group, when thought of as an operation of sorts of some normal subgroup N or its parent group G, preserves several important properties of groups such as commutativity and solvability. Third, the quotient group comes in many forms (for this reason, one might prefer to use the plural terminology, but this is inconsequential). Fourth and finally, the overcoming of the challenge of understanding the basics of the quotient group can snowball and proceed to enlighten students of mathematics further by providing a gateway to comprehension of homomorphisms. With few possible exceptions, exempli gratia the fact that all vectors spaces over a field F with dimension n are isomorphic, the so-called Isomorphism Theorems, which primarily concern themselves with quotient groups and related structures, are the first isomorphisms or even homomorphisms of serious substance a student comes across.

At least with respect to group theory and introductory algebra, I feel a true, comprehensive understanding of quotient groups and homomorphisms can provide a foundation of knowledge and a confidence that can propel a student to further success. In mathematics, perchance algebra especially, one often feels (s)he understands something and is supremely confident in that fact, when, in fact, the material is far deeper than (s)he realizes. The more abstract the content and the simpler the beginnings, the worse the predicament. Therefore, courses in general algebra, category theory, and real analysis provide excellent examples of courses where it is quite easy for students to fall victim to their own hubris and fall behind. As such, many students fail these courses. I believe the status quo described hitherto is the primary reason for this, not the difficulty innate to the subject matter, the abstract nature (at least directly), or other commonly cited attributes of courses such as those previously listed.

To ensure I treat this subject with the care I feel it deserves, I have split the discussion into parts. While I lack the foresight to state now how many parts there will be, I would expect it to be around 3. This will come in eight parts. This series is now concluded and consists of six parts. Thus far, I have yet to have any “real math”, that is no equations, no formulas, no theorems, no lemmas, no corollaries, and so forth. This is somewhat intentional. Those who will benefit most from these posts are students studying mathematics at the undergraduate level, and for many of these students, mathematical maturity and experience is sparse. The idea that mathematics should consist wholly of symbolic manipulations or the theorem-proof-theorem-proof format so ubiquitous in some materials is a misconception worthy of breaking. Terence Tao has an excellent blog post that expounds upon this matter, discussing his idea of how mathematics is neither about loose arguments nor entirely rigorous proof, the likes of which even computers could understand. But, I digress. My point is: this post is mathematical despite its lack of those features considered by many to be essential for anything of mathematical substance. Moreover, mathematics a human endeavor, involving language as well as abstract representations of ideas.

But, for those who still feel they have been cheated by this post’s lack of “real math,” I give you two definitions which will lead us into the second post on this topic, which I hope to post in the next few days.

Definition 1. Let H be a subgroup of G, denoted H < G, and let g \in G, then the set gH=\{gh : h \in H \} is called the left coset of H (in G).

Definition 2. Let H < G. If \forall g \in G we have that gH=Hg, then H is said to be a normal subgroup of G, or simply “H is normal in G.” Normal subgroups are denoted by an left triangle, H \triangleleft G.