Binomial

A binomial is an algebraic expression that consists of two unlike terms, including constants and variables, connected by arithmetic operators such as the plus (+) and minus (-). For example, 2x + 3y is a binomial. Algebraic expressions are classified as monomial (one term), binomial (two terms), and trinomial (three terms) based on the number of terms, as shown in the image below:

A binomial coefficient is a numerical factor that appears in front of each term when expanding an expression like (x + y)². The binomial expression (x + y)ⁿ can be expanded as

(x + y)ⁿ = ⁿC₀ xⁿ y⁰ + ⁿC₁ x^{n - 1} y¹ + ⁿC₂ x^{n - 2} y² + … + ⁿC_{n - 1} x¹ y^{n - 1} + ⁿC_n x⁰ yⁿ.

After expanding (x + y)⁵, we get the expanded form like:

 (x + y)⁵ = ⁵C₀ x⁵ y⁰ + ⁵C₁ x⁴ y¹ + ⁵C₂ x³ y² + ⁵C₃ x² y³ + ⁵C₄ x¹ y⁴ + ⁵C₅ x⁰ y⁵ 

= x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵. 

In this expansion, the numbers 1, 5, 10, 10, 5, 1 are the binomial coefficients. When we arrange these binomial coefficients in a triangle, we will get Pascal’s Triangle. 

In Pascal’s Triangle, each row represents the binomial coefficients for the expression of (x + y)ⁿ, where n corresponds to the row number starting from 0.

Row 0 (n = 0) = (x + y)⁰ = Coefficients: 1
Row 1 (n = 1) = (x + y)¹ = Coefficients: 1, 1
Row 2 (n = 2) = (x + y)² = Coefficients: 1, 2, 1
Row 3 (n = 3) = (x + y)³ = Coefficients: 1, 3, 3, 1
 

Factoring binomials means breaking them into smaller pieces that can be multiplied to get the original expression. There are four methods for factorizing binomials:

• Factoring Binomials Using Greatest Common Factor

• Factoring Binomials Using the Difference of Squares

• Factoring Binomials Using the Sum of Cubes

• Factoring Binomials Using Difference of Cubes

Factoring Binomials Using Greatest Common Factor

Take out the common number or common terms from both terms.
For example, 2x² + 6x, both terms share a common factor 2x, which can be factored out.
So the binomial will become 2x(x + 3). 

Factoring Binomials Using the Difference of Squares

If two terms don’t share a common factor, they can still be factorized if they follow a special pattern.
If a binomial is in the form of a² - b², we can use the identity: a² - b² = (a + b)(a - b).
For example, a² - 9, since 9 is a perfect square (3²), it can be rewritten as, a² - 9 = (a + 3)(a - 3).

Factoring Binomials Using the Sum of Cubes

If we are adding cubes like x³ + 27, here the 27 can be written as 3³.
We can apply the identity: a³ + b³ = (a + b)(a² - ab + b²).
Therefore, x³ + 27 can be written as (x + 3)(x² - 3x + 9).

Factoring Binomials Using the Difference of Cubes

When dealing with the difference of two cubes, such as y³ - 64, we can use the identity, a³ - b³ = (a - b)(a² + ab + b²).
Since 64 is 4³, the expression can be factored as: y³ - 64 = (y - 4)(y² + 4y + 16).
 

A binomial is an expression made up of exactly two terms joined by a plus or a minus sign. (x +3) and (x - 7) are examples of binomials. Squaring a binomial means multiplying the binomial by itself. We can use three identities or formulas for squaring a binomial.

• When both terms are positive, we can use:
  (a + b)² = a² + 2ab + b²
   

• When the second term is negative, use:
  (a - b)² = a² - 2ab + b²
   

• If both terms are negative, the result will still be positive, just like squaring any positive number. So we can use, 
  (-a - b)² = a² + 2ab + b²

Example: Find the square of (3y + 2)

Here, both terms are positive.
So, we can use, (a + b)² = a² + 2ab + b²

(3y + 2)² = (3y)² + 2(3y)(2) + 2²
= 3y² + 12y + 4

Binomials and binomial coefficients are widely used in real life, especially in areas like mathematics, science, and finance. They help to solve problems involving probabilities, patterns, and algebraic expressions. Here are some of the real-life examples where binomials are used.

• Games and Probability: When we are flipping coins, rolling a die, or playing cards, binomial coefficients are used to figure out the chances of getting a certain result. 
  
   
• Business and Finance: In business and finance, binomial models help to predict stocks, estimate future profits, and calculate risks. 
  
   
• Genetics: Binomials are used to calculate the chances of inheriting traits from parents in genetics. 
  
   
• Computer science: Binomial coefficients are used in data compression, algorithms, graphics and animations, and in machine learning, especially in combinations and permutations.