Binomial Expansion Formula

The Binomial Theorem is a quick way to multiply or expand a binomial statement. The intensity of the expressiveness has been amplified significantly. Multiplication of such statements is always difficult with large powers and phrases, as we all know. However, binomial expansions and formulas are extremely helpful in this area. The Binomial Theorem and the Binomial Theorem Formula will be discussed in this article. Let's start with a  few examples to learn the concept.

What is Binomial Expansion, and How does It work?

The binomial theorem is a mathematical expression that describes the extension of a binomial's powers. According to this theorem, the polynomial (x+y)n can be expanded into a series of sums comprising terms of the type an xbyc.

The exponents b and c are non-negative integers, and b + c = n is the condition. In addition, depending on n and b, each term's coefficient is a distinct positive integer.

For n = 4, consider the following:

(x+y)⁴=x⁴+4x³y+6x²y²+4xy³+y⁴

It is self-evident that multiplying such phrases and their expansions by hand would be excruciatingly uncomfortable. Thankfully, someone has devised a formula for this growth, which we can employ with ease.

Formula for the Binomial Theorem

The formula for the Binomial Theorem is written as follows:

\[(x+y)^n=\sum_{k=0}^{n}(nc_r)x^{n-k}y^k\]

Also, remember that n! is the factorial notation. It reflects the product of all whole numbers between 1 and n in this case.

The following are some expansions:

(x+y)1=x+y

(x+y)2=x²+2xy+y²

(x+y)3=x³+3x²y+3xy²+y³

(x+y)n  

Properties of the Binomial Expansion

• In total, there are n+1 terms.

• xn is the initial term, while isyn is the last term.

• The exponent of x declines by 1 from term to term as we progress from the first to the last. While the exponent of y grows by one, the exponent of x grows by one. In addition, the total of both exponents in each term is n.

• We can simply determine the coefficient of the following phrase by multiplying the coefficient of each term by the exponent of x in that term and dividing the product by the number of that term.

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 give 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:

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

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 ‘n’ as {0, 1, 2 …..}, the binomial expansions of (a+b)n for different values of ‘n’ as shown below. 

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. 

Solved Examples 

1. Find the first four terms of the expansion using the binomial series: \[\sqrt[3]{1+x}\]

First, we will write expansion formula for \[(1+x)^3\] as follows:

\[(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+.......\]

Put value of n=\frac{1}{3}, till first four terms:

\[(1+x)^\frac{1}{3}=1+\frac{1}{3}x+\frac{\frac{1}{3}(\frac{1}{3}-1)}{2!}x^2+\frac{\frac{1}{3}(\frac{1}{3}-1)(\frac{1}{3}-2)}{3!}x^3\]

Thus expansion is:

\[(1+x)^\frac{1}{3}=1+\frac{1}{3}x-\frac{x^2}{9}+\frac{5x^3}{81}\]