Binomials are mathematical expressions that contain two members or terms, whether these are numbers or abstract representations that generalize a finite or infinite number of numbers. Binomials are, therefore, compositions of two terms.
In mathematical language, a term is understood to be an operational unit separated from another by an addition (+) or subtraction (−) sign. Combinations of expressions separated by other mathematical operators do not fall into this category.
Square binomials ( or square binomials) are those in which the sum or subtraction of two terms must be raised to the power of two.
An important fact about exponentiation is that the sum of two squared numbers is not equal to the sum of the squares of those two numbers, but rather 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 and Pascal to develop 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 exponentiation of binomials is carried out, and this was expressed in mathematical language (although it can be explained in words),
- The second showed in a much more didactic way how the coefficients of the developments of powers increase as the exponent to which the expression is raised increases.
Newton’s theorem , which like every mathematical theorem has a demonstration, shows that the expansion of (A+B) N has N+1 terms, of which the powers of A start with N as exponent in the first and decrease 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’s triangles is usually applied.
With this 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
See also: Algebraic language
Examples of solving 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 + 3/5 AB + ¼ B 2
- (2A 2+ 5B 2 ) 2 = 4A 4 + 20 A 2 B 2 + 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
