Binomial is a polynomial with only terms. For example, x + 2 is a binomial, where x and 2 are two separate terms. Also, the coefficient of x is 1, the exponent of x is 1 and 2 is the constant here. Therefore, A binomial is a two-term algebraic expression that contains variable, coefficient, exponents and constant. 

Another example of a binomial polynomial is x2 + 4x. Thus, based on this binomial we can say the following:

• x² and 4x are the two terms
• Variable = x
• The exponent of x² is 2 and x is 1
• Coefficient of x² is 1 and of x is 4

Is 2x a binomial? 2x has only one term, hence it is not a binomial.

Binomial Definition

The algebraic expression which contains only two terms is called binomial. It is a two-term polynomial. Also, it is called a sum or difference between two or more monomials. It is the simplest form of a polynomial.

• When expressed as a single indeterminate, a binomial can be expressed as;

ax^m + bxⁿ

Where a and b are the numbers, and m and n are non-negative distinct integers. x takes the form of indeterminate or a variable.

• In Laurent polynomials, binomials are expressed in the same manner, but the only difference is the exponents, m and n can be negative. Therefore, we can write it as;

ax^-m + bx^-n

Examples of Binomial

Some of the binomial examples are;

• 4x²+5y²
• xy²+xy
• 0.75x+10y²
• x+y
• x² + 3
• x³ + 2x
• 9 + 7y
• m + 2n

Other Polynomials

Apart from the binomial, the other two types of the polynomial are:

• Monomial
• Trinomial

Monomial: When an expression is having only one term or a single term, then such polynomial is known as a monomial.  Examples of monomial are 3x, 4, 5x², 6x³, etc.

Trinomial: A trinomial is a polynomial that has only three terms. For example, x² – 3 + 3x.

Binomial Equation

Any equation that contains one or more binomial is known as a binomial equation.

Some of the examples of this equation are:

x² + 2xy + y² = 0

v = u+ 1/2 at²

Operations on Binomials

There are few basic operations that can be carried out on this two-term polynomials are:

• Factorisation
• Addition
• Subtraction
• Multiplication
• Raising to n^th Power
• Converting to lower-order binomials

Factorization of Binomials

We can factorise and express a binomial as a product of the other two.
For example, x² – y² can be expressed as (x+y)(x-y).

Addition of Binomials

Addition of two binomials is done only when it contains like terms. This means that it should have the same variable and the same exponent.
For example,
Let us consider, two equations.
10x³ + 4y and 9x³ + 6y
Adding both the equation = (10x³ + 4y) + (9x³ + 6y)
Therefore, the resultant equation = 19x³ + 10y

Subtraction of Binomials

Subtraction of two binomials is similar to the addition operation as if and only if it contains like terms.
For example,
12x³ + 4y and 9x³ + 10y
Subtracting the above polynomials, we get;

(12x³ + 4y) – (9x³ + 10y)
= 12x³ + 4y – 9x³ – 10y
Therefore, the resultant equation is = 3x³ – 6y

Multiplication of Binomials

When multiplying two binomials, the distributive property is used and it ends up with four terms. It is generally referred to as the FOIL method. Because in this method multiplication is carried out by multiplying each term of the first factor to the second factor. So, in the end, multiplication of two two-term polynomials is expressed as a trinomial.

For example, (mx+n)(ax+b) can be expressed as max²+(mb+an)x+nb

Raising to nth Power

A binomial can be raised to the nth power and expressed in the form of;

(x + y)ⁿ

Converting to lower-order binomials

Any higher-order binomials can be factored down to lower-order binomials such as cubes can be factored down to products of squares and another monomial.

For example, x³ + y³ can be expressed as (x+y)(x²-xy+y²)

Binomial Expansion

In Algebra, binomial theorem defines the algebraic expansion of the term (x + y)ⁿ. It defines power in the form of ax^by^c. The exponents b and c are non-negative distinct integers and b+c = n and the coefficient ‘a’ of each term is a positive integer and the value depends on ‘n’ and ‘b’.

For example, for n=4, the expansion (x + y)⁴ can be expressed as

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

The coefficients of the binomials in this expansion 1,4,6,4, and 1 forms the 5th degree of Pascal’s triangle.

The general theorem for the expansion of (x + y)ⁿ is given as;

(x + y)ⁿ = \({n \choose 0}x^{n}y^{0}\)+\({n \choose 1}x^{n-1}y^{1}\)+\({n \choose 2}x^{n-2}y^{2}\)+\(\cdots \)+\({n \choose n-1}x^{1}y^{n-1}\)+\({n \choose n}x^{0}y^{n}\)

Some of the methods used for the expansion of binomials are :

• Pascal’s Triangle
• Factorials
• Combinations
• Binomial Series

Binomial Formula

Binomial Distribution

The term binomial distribution is used for a discrete probability distribution. There are only two outcomes here: Success and Failure. Learn in detail Binomial distribution and binomial distribution formula at BYJU’S.

Problems and Solutions

Question 1: Find the binomial from the following terms?

1. 5x + 6y
2. 5 y
3. 6xy
4. 6x ÷ y

Solution:

1. Option 1: 5x + 6y: Here, addition operation makes the two terms from the polynomial
2. Option 2: 5 * y: Multiplication operation produces 5y as a single term
3. Option 3: 6xy: Multiplication operation produces the polynomial 6xy as a single term
4. Option 4: 6x÷ y: Division operation makes the polynomial as a single term. 

Therefore, the solution is 5x + 6y, is a binomial that has two terms.

Question 2: Multiply (5 + 4x) . (3 + 2x).

Solution: (5 + 4x)(3 + 2x)

= (5)(3) + (5)(2x) + (4x)(3) + (4x)(2x)

= 15 + 10x + 12x + 8(x)² 

= 15 + 22x + 8x²

Question 3: Add: 6a + 8b – 7c, 2b + c – 4a and a – 3b – 2c.

Solution: (6a + 8b – 7c) + (2b + c – 4a) + (a – 3b – 2c) 

= 6a + 8b – 7c + 2b + c – 4a + a – 3b – 2c

Arranging the like terms, we get;

= 6a – 4a + a + 8b + 2b – 3b – 7c + c – 2c

= 3a + 7b – 8c

Question 4:Add 2x² + 6x and 3x² – 2x.

Solution: 2x² + 6x   +   3x² − 2x

Arrange the like terms

2x² + 3x² + 6x − 2x

(2+3)x² + (6−2)x 

5x² + 4x 

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