What are the English names for the rules for expanding a sum or a difference squared? $(a+b)^2=a^2+2ab+b^2$
$(a-b)^2=a^2-2ab+b^2$
What are these two called in English? Are they called anything at all? I looked around on some math websites and Wikipedia, but I didn't find these rules. I did find the conjugate rule for the difference of two squares. That was useful, and I have now memorized that name, but that was not exactly what I was looking for.
Maybe they are not considered to be noteworthy rules so they have not been given a name in the English speaking world?
Update:
These are called "kvadreringsregler" in Swedish. I have found some translations to other languages, but not to English.


*

*Danish: Kvadratsætningen

*Swedish: Kvadreringsreglerna

*Norwegian: Kvadratsetningene

*Polish: Wzory skróconego mnożenia

*French: Identité remarquable

*English: ???


The proposed "binomial theorem" is called "binomialsatsen" in Swedish. This is considered to be a more generalized form of "kvadreringsregelerna". The French name for this literally translates into "remarkable identities". That gave me some interesting web search results.

We call remarkable identities to [sic] some binomial products that appear
  very often in calculations with algebraic expressions.

Source: Mathspadilla
Could this be it?
 A: I call both those identities binomial squares in my classes.
A: Here in Spanish they are examples of "productos notables" ("remarkable products" might be the closest translation). The list includes:
$\begin{align}
(a + b)^2 &= a^2 + 2 a b + b^2 \\
(a - b)^2 &= a^2 - 2 a b + b^2 \\
(a + b) (a - b) &= a^2 - b^2
\end{align}$
A: Identité remarquable!
In other words, remarkable identities!

We call remarkable identities to [sic] some binomial products that appear
  very often in calculations with algebraic expressions.

Source: Mathspadilla
Square of an addition:
$(a + b)^2 = a^2 + 2ab + b^2$
Square of a subtraction:
$(a - b)^2 = a^2 - 2ab + b^2$
A: I will use the information I have gathered so far to try to provide proper English translations. I will be using Swedish as the base language.
Kvadreringsreglerna

Första kvadreringsregeln:
$(a+b)^2=a^2+2ab+b^2$
This is often referred to as the first rule. 
Andra kvadreringsregeln:
$(a-b)^2=a^2-2ab+b^2$
This is referred to as the second rule.
The first thing I realize is that there are no specific names for these in English. So I would propose the translation identities of binomial squares for the two of them. I would then suggest the name identity of binomial squares of a sum for the first rule, and identity of binomial squares of a difference for the second rule.
When used in a sentence:


*

*Använd kvadreringsreglerna för att utveckla uttrycket.

*Use the identities of binomial squares to expand the expression.

*Använd kvadreringsreglerna för att faktorisera.

*Use the identities of binomial squares to factorize.
What we refer to as rules ("regler") in Swedish, in this context, is referred to as identities in English. We do have the term "identitet" but we usually don't use it in this context. These identities are so important that we have decided to give them a special name. Apparently we are not alone in this, there are several other languages that have a special name for these identities.


*

*French: Identité remarquable

*Spanish: Productos notables

*Portuguese: Produtos notáveis

*Dutch: Merkwoardig product

*Italian: Prodotto notevole


Dare we translate that into English as "notable products"? Or as "remarkable identities" I suggested earlier? Most of the sources I found that use those terms in English are web sites that are authored by people from India, Spain or Portugal. They may not be native English speakers, so they translate it literally from their own language.
For what I have seen so far, it's only in Nordic countries that these are referred to as rules or theorems.


*

*Swedish: Kvadreringsreglerna (lit. squering rules)

*Danish: Kvadratsætningen (square theorem)

*Norwegian: Kvadratsetningene (square theorem)
This is really an example of the binomial theorem. But the binomial theorem is considered to be the more generalized, all-encompassing theorem for higher powers. When dealing with binomials with powers of 2, we call for the "kvadreringsregler" (Swedish), or "Kvadratsetningene" (Norwegian). For powers higher than 2 (or 3 in case of Swedish) we call for the "binomialsatsen" (binomial theorem).
Kuberingsregeln

Even though the question didn't call for cubes, but I might as well add these in.
Första kuberingsregeln:
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
Andra kuberingsregeln:
$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
Similarly, these rules or identities apply to binomials of third power. I would propose the name identities of binomial cubes for the two of them, identity of binomial cube of a sum for the first rule, and identity of binomial cube of a difference for the second rule.
They might also be referred to as binomial theorem for cubes. Similarly, one might use a similar name for the powers of 2, by only replacing the word "cubes" with "squares". That might provide a more elegant description, without going into detail of "sum" and "difference".
Konjugatregeln

Again, the question didn't call for this one. But I will add this in, since it is one of those "notable product" identities used in other languages.
$a^2 - b^2 = (a+b)(a-b)$
This one actually does have an established English name. It's called the difference of two squares. The Swedish name for it might be translated as conjugate identity of two squares. One might use the "conjugate identity of two squares" to expand or to factorize an expression.
Algebraic identities

These are all collectively referred to as algebraic identities in English.
A: I note in your longer answer to yourself that you skip over a subtle but important distinction: In Swedish, and the other Nordic tongues, the names you give refer to the formulae as a whole: the "squaring rules", the "cubing rules", the "conjugate rule". In English, the only one "named" is the conjugate rule, but that name does not actually refer to the formula. 
Instead, it refers to one of the expressions, specifically the produced (I guess? neither "unfactored" nor "multiplied" feel right to me here) one, and this is why it has a name: the difference of two squares is easy to name as an English phrase (or a phrase in any natural language, really) while the two squaring rules produce trinomials which are harder to name without invoking symbols. If I were to try, it would be "the sum of two squares with twice the product of their roots" for the first squaring rule and "the difference between the sum of two squares and twice the product of their roots" for the second. The cubing rules, meanwhile, produce quadrinomials, which are even more of a nightmare to name.
Now, English could just as easily use the name for the factored expression for the formulae: "square of a sum", "square of a difference", "cube of a sum", "cube of a difference" and "conjugate product" (or, for the uninitiated, "the product of the sum of two values with their difference"), but it doesn't. But then, neither does it use "difference of two squares" for the conjugate rule, as you claim. As a native English speaker (General American dialect; results may vary), I never think "$a^2-b^2=$ $(a+b)(a-b)$" when I hear or say "difference of two squares". I think "$a^2-b^2$ which equals $(a+b)(a-b)$". The two expressions are equivalent mathematically but not syntactically and are separate in my mind.
Personally, I like the idea of a literal translation from the Swedish, with perhaps the modifier "binomial" for the squaring and cubing rules. They're simple and descriptive, perfect for teaching. 
