Binomial Expansion Formula

Binomial Theorem General Term

Binomial Expansion is one of the methods used to expand the binomials with powers in algebraic expressions. In algebra, a binomial is an algebraic expression with exactly two terms (the prefix ‘bi’ refers to the number 2). If a binomial expression (x + y)n is to be expanded, a binomial expansion formula can be used to express this in terms of the simpler expressions of the form ax + by + c in which ‘b’ and ‘c’ are non negative integers. The value of ‘a’ completely depends on the value of ‘n’ and ‘b’. This section gives a deeper understanding of what is the general term of binomial expansion and how binomial expansion is related to Pascal's triangle. 


Mathematical Form of the General Term of Binomial Expansion

Any binomial of the form (a + x) can be expanded when raised to any power say ‘n’ using the binomial expansion formula given below.

( a + x )n = an + nan-1x + \[\frac{n(n-1)}{2}\] an-2 x2 + …. + xn

The above stated formula is more favorable when the value of ‘x’ is much smaller than that of ‘a’. This is because, in such cases, the first few terms of the expansions gives a better approximation of the expression’s value. The expansion always has (n + 1) terms. The general term of binomial expansion can also be written as:

(a + x)n = \[\sum_{k=0}^{n}\frac{n!}{(n-k)!k!}a^{n-k}x^{k}\]

Note that the factorial is given by

N! = 1 . 2 . 3 … n

0! = 1


Important Terms Involved in Binomial Expansion

The expansion of a binomial raised to some power is given by the binomial theorem. It is most commonly known as Binomial expansion. Various terms used in Binomial expansion include:

  • General term

  • Middle term

  • Independent term

  • To determine a particular term

  • Numerically greatest term

  • Ratio of consecutive terms also known as the coefficients


Binomial Theorem and Pascal’s Triangle:

Pascal’s triangle is a triangular pattern of numbers formulated by Blaise Pascal. The binomial expansion of terms can be represented using Pascal's triangle. To understand how to do it, let us take an example of a binomial (a + b) which is raised to the power ‘n’ and let ‘n’ be any whole number. For assigning the values of to ‘n’ as {0, 1, 2 …..}, the binomial expansions of (a+b)n for different values of ‘n’ as shown below. 

(a + b)0 =

(a + b)1 =

(a + b)2 =

(a + b)3 =

(a + b)4 =

(a + b)5 =

1

a + b

a2 + 2ab + b2

a3 +3a2b + 3ab2 + b3

a4 + 4a3b + 6a2b2 + 4ab3 + b4

a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5


With this kind of representation, the following observations are to be made.

  • Each expansion has one term more than the chosen value of ‘n’.

  • In each term of the expansion, the sum of the powers is equal to the initial value of ‘n’ chosen.

  • The powers of ‘a’ start with the chosen value of ‘n’ and decreases to zero across the terms in expansion whereas the powers of ‘b’ start with zero and attains value of ‘n’ which is the maximum.

  • The coefficients start with 1, increase till half way and decrease by the same amounts to end with one. 


Properties of Binomial Theorem

There are numerous properties of binomial theorems which are useful in Mathematical calculations. The few important properties of binomial coefficients are:

  • Every binomial expansion has one term more than the number indicated as the power on the binomial. 

  • Exponents of each term in the expansion if added gives the sum equal to the power on the binomial.

  • The powers of the first term in the binomial decreases by 1 with each successive term in the expansion and the powers on the second term increases by 1. 

  • It is important to note that the coefficients form a symmetrical pattern. 


Binomial Expansion Formula Practical Applications

Binomial expansions are used in various mathematical and scientific calculations that are mostly related to various topics including 

  • Kinematic and gravitational time dilation

  • Kinetic energy

  • Electric quadrupole pole

  • Determining the relativity factor gamma


Are Algebraic Identities Connected with Binomial Expansion?

The answer to this question is a big YES!! A few algebraic identities can be derived or proved with the help of Binomial expansion. The following identities can be proved with the help of binomial theorem.

  • (x + y)2 = x2 + 2xy + y2

  • (x - y)2 = x2 - 2xy + y2

  • (x + y)3 = x3 + 3x2y + 3xy2 + y3 

  • (x - y)3 = x3 - 3x2y + 3xy2 - y3 


Binomial Expansion Example Problems

  1. Evaluate (3 + 7)3 Using Binomial Theorem. 

Solution:

The binomial expansion formula is given as:

(x+y)n = xn + nxn-1y + n(n−1)2! xn-2y2 +…….+ yn

In the given problem,

x = 3 ; y = 7 ; n = 3

(3 + 7)3 = 33 + 3 x 32 x 7 + (3 x 2)/2! x 31 x 72 + 73

= 27 + 189 + 441 + 343

(3 + 7)3 = 1000


Fun Facts

  • The number of terms in a binomial expansion of a binomial expression raised to some power is one more than the power of the binomial expansion. 

  • Isaac Newton takes the pride of formulating the general binomial expansion formula. 

  • Binomial theorem can also be represented as a never ending equilateral triangle of algebraic expressions called the Pascal’s triangle. 

FAQ (Frequently Asked Questions)

1. What is the Binomial Expansion Formula?

Binomial expansion is a method used in algebra to expand a binomial algebraic expression. Binomial is an expression with two different terms. The method is also popularly known as Binomial theorem. Basically, binomial theorem demonstrates the sequence followed by any Mathematical calculation that involves the multiplication of a binomial by itself as many times as required. The general formula for binomial expansion is given as follows:

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2. Where is the Binomial Theorem Used?

Binomial theorem is used as one of the quick ways of expanding or obtaining the product of a binomial expression raised to a specified power (the power can be any whole number). It is used in all Mathematical and scientific computations which involve these types of calculations. A few concepts in Physics that use Binomial expansion formula quite often are:

  • Kinematic and gravitational time dilation

  • Kinetic energy

  • Electric quadrupole pole

  • Determining the relativity factor gamma