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.
This is often referred to as the first rule.
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).
Even though the question didn't call for cubes, but I might as well add these in.
$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
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".
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.
These are all collectively referred to as algebraic identities in English.