Binomial

In algebra, a binomial is an expression that has two unlike terms connected through an addition or subtraction operator in between. For example, 2xy + 7y is a binomial since there are two terms. Algebraic expressions can be categorized into different types depending upon the number of terms present, like monomial, binomial, trinomial, etc.

In this article, we will explore the binomial expression in algebra, its properties and its identities that are used to solve various problems in algebra. We shall go through different solved examples based on binomial for a better understanding of the concept.

A binomial is an algebraic expression that has two terms. In other words, an algebraic expression consisting of two unlike terms having constants and variables is a binomial expression. These terms are joined using arithmetic operators such as + (plus) and –(minus). A binomial, along with monomial, trinomial, quadrinomial, etc is categorized under algebraic expressions based on the number of terms it contains. Observe the following unlike terms mentioned in the image below. These are a few illustrations explaining what and how exactly polynomials are categorized as binomial, monomial, and trinomial.

Binomial is an algebraic expression that contains two different terms connected by addition or subtraction. In other words, we can say that two distinct monomials of different degrees connected by plus or minus signs form a binomial. For example, consider two monomials, 2x and 5x¹⁰. The expression to add these monomials gives a binomial given by, 2x + 5x¹⁰.

Binomial coefficients are the positive integers that are the coefficients of terms in a binomial expansion. We know that a binomial expansion '(x + y) raised to n' or (x + n)ⁿ can be expanded as, (x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n-1 x¹y^n-1 + ⁿC_n x⁰yⁿ, where, n ≥ 0 is an integer and each ⁿC_k is a positive integer known as a binomial coefficient using the binomial theorem. Here, n is the power of the binomial, and k is 1 less than the number of the term we are considering where n ≥ k ≥ 0. The formula to find the binomial coefficient of the k^th term of any binomial raised to power n is given below,

ⁿC_k = (n!) / [k ! (n-k)!]

Let us take an example to understand it better. To find the binomial coefficients of the expansion (x + 4)⁵, let us apply the above binomial coefficient formula. Here, the value of n is 5.

For k = 0 (for the first term), we have, ⁵C₀ = (5!) / [0! (5-0)!]

⇒ ⁵C₀ = 5!/5! = 1.

It implies that the binomial coefficient of the first term of the expression (x + 4)⁵ is 1. Similarly, for k=1 (for the second term), we have, ⁵C₁ = (5!) / [1! (5-1)!].

⇒ ⁵C₁ = 5!/4! = (5 × 4!)/4!

= 5

It means the binomial coefficient of the second term of the expression (x + 4)⁵ is 5. Similarly,

When k=2, ⁵C₂ = (5!) / [2! (5-2)!]

⇒ ⁵C₂ = 5!/(2! × 3!) = (5 × 4 × 3!)/(2 × 1 × 3!)

⇒ ⁵C₂ = (5 × 4)/2 = 10

When k=3, ⁵C₃ = (5!) / [3! (5-3)!]

⇒ ⁵C₃ = 5!/(3! × 2!) = (5 × 4 × 3!)/(3! × 2 × 1)

⇒ ⁵C₃ = (5 × 4)/2 = 10

When k=4, ⁵C₄ = (5!) / [4! (5-4)!]

⇒ ⁵C₄ = 5!/(4! × 1!) = (5 × 4!)/(4!)

⇒ ⁵C₄ = 5

When k=5, ⁵C₅ = (5!) / [5! (5-5)!]

⇒ ⁵C₅ = 5!/(5! × 0!) = 5!/5!

⇒ ⁵C₅ = 1

Using this binomial coefficients formula, we can find out the coefficients of the terms without even expanding the expansion. The coefficients of all the 6 terms of the binomial (x + 4)⁵ are 1, 5, 10, 10, 5, and 1. One interesting fact here is that if we find and arrange the binomial coefficients of the expansion in the triangle form, we will get a special type of triangle known as Pascal's triangle.

Factoring is the process of expressing an algebraic expression as a product of its factors. Factoring binomial means breaking down the binomial into the product of two expressions. As we know that binomials are expressions containing two terms, so by factoring a binomial, we will get its two factors of a lower degree. There are four rules of factoring binomial which are given below:

Rule 1: Factoring Binomial by using the greatest common factor (GCF).

If both the terms of the given binomial have a common factor, then it can be used to factor the binomial. For example, in 2x² + 6x, both the terms have a greatest common factor of 2x.

When 2x² ÷ 2x = x and, 6x ÷ 2x = 3

Therefore, 2x² + 6x can be factored as 2x(x + 3).

Rule 2: Factoring Binomial by using the difference of squares.

With some binomials, there are no common factors of both the terms, but still, we can factorize them. One such way is by considering the difference of squares. If we recognize that both the terms are in the form of x² - y², then, we can use the following identity to factorize such binomials: x² - y² = (x+y)(x-y). For example, let us factorize a² - 9. Here, a² is the square of a, and 9 is the square of 3.

⇒ a² - 9 = a² - 3²

By using the algebraic identity: x² - y² = (x+y)(x-y), we can write it as (a+3)(a-3). Therefore, a² - 9 = (a+3)(a-3).

Rule 3: Factoring Binomial by using the sum of cubes.

Sometimes, binomials are given as the sum of cubes, for example, x³ + 27. In such cases, the following algebraic identity can be used to factorize the binomial: a³ + b³ = (a + b)(a² - ab + b²). For example, let us factorize the binomial x³ + 27. Here x³ is the cube of x and 27 is the cube of 3.

⇒ x³ + 27 = x³ + 3³

By using the algebraic identity: a³ + b³ = (a + b)(a² - ab + b²), we can write it as (x + 3)(x² - 3x + 9). Therefore, x³ + 27 = (x + 3)(x² - 3x + 9).

Rule 4: Factoring Binomial by using the difference of cubes identity.

Another type of binomial is the difference of cubes, for example, y³ - 64. In such cases, the following algebraic identity can be used to factorize the binomial: a³ - b³ = (a - b)(a² + ab + b²). For example, let us factorize the binomial y³ - 64. Here y³ is the cube of y and 64 is the cube of 4.

⇒ y³ - 64 = y³ - 4³

By using the algebraic identity: a³ - b³ = (a - b)(a² + ab + b²), we can write it as (y - 4)(y² + 4y + 16). Therefore, y³ - 64 = (y - 4)(y² + 4y + 16).

Here, it is important to note that every binomial cannot be factored into two expressions containing rational coefficients. It should be either of the four types explained above. One example of a binomial that cannot be factored is 3a² + 16.

The square of a binomial is the sum of the square of the first term, twice the product of both terms, and the square of the second term. When the sign of both terms is positive, then we use the following identity for squaring binomial: (a + b)² = a² + 2ab + b². When the sign of the second term is negative, then we use the following identity: (a - b)² = a² - 2ab + b². And, when both the terms are negative, then the following identity is to be used for squaring binomial: (- a - b)² = a² + 2ab + b². Let us take an example to understand the concept of squaring binomial. Find the square of 2x - 5.

⇒ (2x - 5)² = (2x)² - 2 × (2x) × 5 + (5)² [By using the identity: (a - b)² = a² - 2ab + b²]

= 4x² - 20x + 25

Therefore, (2x - 5)² = 4x² - 20x + 25. Here, it is important to note that the square of a binomial is always a trinomial.

Important Notes on Binomial

• A binomial is an algebraic expression consisting of two different monomials of different degrees connected by the + or – sign.
• We can factorize binomial expressions using different rules of factoring.
• To find the value of binomial expression raised to a power, we use the binomial theorem.

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What is a Binomial in Algebra?

A binomial is an algebraic expression having exactly two unlike terms, including the variables and the constant. For example, 2x + 3, 3x + 4y, etc. In other words, we can say that two distinct monomials connected by plus or minus signs give a binomial expression.

What is an Example of a Binomial?

One example of a binomial is x + 2. Any polynomial having exactly two terms connected through a plus or minus sign in between is an example of binomial. Some other examples of binomials are 5x³ - 9, 4xy - 9pq, 5 - 2m, etc.

What Type of Expression is a Binomial?

An algebraic expression that contains two unlike terms is called a binomial. In simple words, it is an algebraic expression that has two non-zero terms. Let us see the examples of a binomial expression: 2a + b is a binomial in two variables a and b. Few more examples are 2xy + 3x, 7 + 9y, etc.

Is x² + 2 a Binomial?

Yes, x² + 2 is a binomial. We know that any algebraic expression with two unlike terms is considered binomial. In the stated expression x² + 2, we have two non-zero terms joined together by an addition sign in between.

What is Binomial Theorem?

The binomial theorem is an equation or a formula to find any power of a binomial without actually multiplying it. The binomial theorem of (x + y) raised to any power 'n' is given by, (x + y)ⁿ = (x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n.

What is Binomial Coefficient?

Binomial coefficients are the positive integers attached with each term in a binomial theorem. For example, the expanded form of (x + y)² is x² + 2xy + y². Here, the binomial coefficients are 1, 2, and 1. These coefficients depend on the exponent of the binomial, which can be arranged in a triangle pattern known as Pascal's triangle.

Can a Binomial be a Polynomial?

Yes, a binomial is a type of polynomial, having two non-zero terms. In other words, a polynomial containing two terms is called a binomial.

What is the Degree of a Binomial?

The degree of a binomial is the sum of the exponents of all the variables (written as products) in a term. For example, in the binomial 1 + 2x²y³, the exponent of x is 2, and the exponent of y is 3. So, the degree of this binomial is 5 "(2+3=5)".

What is the Difference Between Monomial, Binomial, Trinomial?

Monomials, binomials, and trinomials are all named according to the number of terms they have. Monomials have a single term, binomials have two terms and trinomials have three terms. For example:

• Monomial: 224x, 8x², 2x⁴
• Binomial:12x + 2y, or, 14y + x³.
• Trinomial: 7x + 14x² + 7x³.

How to Factor a Binomial?

Factoring a binomial means to break it down into the product of two factors. There are four rules or identities used to factor a binomial which are given below:

• ax + ay = a(x + y)
• x² - y² = (x+y)(x-y)
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

How to Square a Binomial?

Squaring a binomial means any binomial raised to a power of 2. The two identities used for squaring binomial are given below:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

How to Cube a Binomial?

Cube of a binomial means finding the third exponent of the given binomial. The identities used to find the cube of a binomial are given below:

• (a + b)³ = a³ + b³ + 3ab (a + b)
• (a - b)³ = a³ - b³ - 3ab (a - b)

What is Binomial Expansion?

A binomial expansion is a formula that is used to find the value of binomial expression raised to a power which is given by, (x + y)ⁿ = (x+y)ⁿ = ⁿC₀ xⁿy⁰ + ⁿC₁ x^n-1y¹ + ⁿC₂ x^n-2 y² + ... + ⁿC_n.