The **pairs** are mathematical expressions in which two members or terms appear, whether these numbers or abstract representations that generalize a finite or infinite amount of numbers. The binomials are thus **two-term compositions** .

In mathematical language, the *term* is understood as the operational unit that is separated from another by a sign of addition (+) or subtraction (-). Those combinations of expressions separated by other mathematical operators do not correspond to this category.

The **square pairs** (or pairs to the square) are __those in which the addition or subtraction of two terms should be raised to the power two__ . An important fact of the potentiation is that the sum of two squared numbers is not equal to the sum of the squares of those two numbers, but that one more term must be added that includes twice the product of A and B.

Precisely this is what motivated **Newton** and **Pascal** to elaborate two considerations that are very useful in understanding the dynamics of these powers: Newton’s theorem and Pascal’s triangles:

- The first one aimed to establish the formula under which the potentiation of the binomials is carried out, and this was expressed in mathematical language (although it can be explained in words),
- The second one showed in a much more didactic way how the coefficients of the power developments are increasing as the exponent to which the expression is raised increases.

The **theorem Newton** , that as every mathematical theorem is a demonstration shows that the development of (A + B) ^{N} has N + 1 terms, of which the powers of A beginning with N as an exponent in the first and will decline to 0 in the last, while the powers of B start with exponent 0 in the first and increase to N in the last: with this it can be said that in each of the terms the sum of the exponents is N.

As for the coefficients, it can be said that the coefficient of the first term is one and that of the second is N, and to determine a coefficient value the theory of Pascal triangles is usually applied.

With that said, it is enough to understand that **the generalization of the square of the binomial works as follows:**

**(A + B) **^{2}** = A **^{2}** + 2 * A * B + B **^{2}

### Examples resolutions of square binomials

**(X + 1)**^{2}**= X**^{2}**+ 2X + 1****(X-1)**^{2}**= X**^{2}**– 2X + 1****(3 + 6)**^{2}**= 81****(4B + 3C)**^{2}**= 16B**^{2}**+ 24BC + 9C**^{2}**(56-36)**^{2}**= 400****(3/5 A + ½ B)**^{2}**= 9/25 A**^{2}**+ ¼ B**^{2}**(2 * A**^{2}**+ 5 * B**^{2}**)**^{2}**= 4A**^{4}**+ 25B**^{4}**(10000-1000)**^{2}**= 9000**^{2}**(2A – 3B)**^{2}**= 4A**^{2}**– 12AB + 9B**^{2}**(5ABC-5BCD)**^{2}**= 25A**^{2}**– 25D**^{2}**(999-666)**^{2}**= 333**^{2}**(A-6)**^{2}**= A**^{2}**– 12A +36****(8a2b + 7ab6y²) ² = 64a4b² + 112a3b7y² + 49a²b12y4****(A**^{3}**+ 4B**^{2}**)**^{2}**= A**^{6}**+ 8A**^{3}**B**^{2}**+ 16A**^{4}**(1.5xy² + 2.5xy) ² = 2.25 x²y4 + 7.5x³y³ + 6.25x4y²****(3x – 4)**^{2}**= 9x**^{2}**– 24x – 16****(x – 5)**^{2}**= x**^{2}**-10x + 25****– (x – 3)**^{2}**= -x**^{2}**+ 6x-9****(3x**^{5}**+ 8)**^{2}**= 9x**^{10}**+ 48x**^{5}**+ 64**