The **binomials** They are mathematical expressions in which two members or terms appear, either these numbers or abstract representations that generalize a finite or infinite quantity of numbers. The binomials are thus compositions of two terms.

In mathematical language, it is understood by ** finished** the operational unit that is separated from another by an addition (+) or subtraction (-) sign. Combinations of expressions separated by other mathematical operators do not fall into this category.

The **square binomials** (or binomials squared) are those in which the addition or subtraction of two terms must be raised to the power two. An important fact about empowerment 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 that includes twice the product of A and B. For example :* (X + 1) ^{2} = X^{2} + 2X + 1, (3 + 6)^{2} = 81, (56-36)^{2} = 400*.

This is precisely what motivated **Newton** already **Pascal** to elaborate two considerations that are very useful when it comes to 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 potentiation of the binomials is carried out, and this was expressed in mathematical language (although it can well be explained with words),
- The second 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 raised increases.

The **Newton’s theorem**, which like every mathematical theorem has a proof, shows that the expansion of (A + B)^{N }has N + 1 terms, of which the powers of A begin with N as an exponent in the first and decrease to 0 in the last, while the powers of B begin with an exponent of 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.

Regarding 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’s triangles is usually applied.

With what has been 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 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
- (TO
^{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