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, therefore, **compositions of two terms** .

In mathematical language, the *term* operational unit is understood to be separated from another by an addition (+) or subtraction (-) sign. 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 enhancement is that the sum of two squared numbers is not equal to the sum of the squares of those two numbers, but one more term must also be added, which includes twice the product of A and B.

This is precisely 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 of them aimed to establish the formula under which the enhancement of the binomials is carried out, and this was expressed in mathematical language (although it may well be explained in words),
- The second one showed in a much more didactic way how the coefficients of the developments of the powers increase as the exponent to which the expression is elevated 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, the coefficient of the first term can be said to be one and that of the second is N, and Pascal’s theory of triangles is usually applied to determine a coefficient value.

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

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

### Examples of square binomial resolutions

**(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**