# Binomial Expansion: Definition, Formulas, Applications, Examples and Binomial Theorem

The algebraic expression which includes only two terms is known as the binomial equation. It is also known as a two-term polynomial. It is the most simplistic form of a polynomial. For example, y + 9 is a binomial expression, where y and 9 are two separate terms. Moreover, the coefficient of y is equal to 1 and the exponent of y is 1 and 9 is the constant in the equation.

There is a set of algebraic identities to determine the expansion when a binomial is raised to exponents two and three. For example:

\(\left(a+b\right)^2=a^2+2ab+b^2. \)

\(\left(a+b\right)^3=\left(a^2+2ab+b^2\right)\left(a+b\right)=a^3+3a^2b+3ab^2+b^3\)

But what if the exponent or the number raised to is bigger? It will become a tedious process to obtain the expansion manually. The binomial expansion formula can simplify this method. Through this article on binomial expansion learn about the binomial theorem with definition, expansion formula, examples and more.

**Binomial Expansion Formula**

The first remark of the binomial theorem was in the 4th century BC by the renowned Greek mathematician Euclids. The binomial theorem states the principle for extending the algebraic expression \( (x+y)^{n}\) and expresses it as a summation of the terms including the individual exponents of variables x and y. Every term in a binomial expansion is linked with a numeric value which is termed a coefficient.

The binomial expansion formula includes binomial coefficients which are of the form \(\left(_k^n\right)\text{ or }\left(^nC_k\right)\)

and it is measured by applying the formula \(\left(^nC_k\right)=\frac{n!}{\left[\left(n-k\right)!k!\right]}\).

The binomial expansion formula is also acknowledged as the binomial theorem formula. Check out the binomial formulas.

Learn more about probability with this article.

When the powers are a natural number:

\(\left(x+y\right)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+^nC_2x^{n-2}y^2+\cdots\cdots+^nC_nx^0y^n\)

OR

\(\left(x+y\right)^n=x^n+nx^{n-1}y+\frac{n\left(n-1\right)}{2!}x^{n-2}y^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^{n-3}y^3+\cdots+y^n\)

When the powers are rational numbers:

Assign n as a rational number and x to be a real number such that | x | < 1 Then:

\(\left(1+x\right)^n=1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+\cdots+\frac{n\left(n-1\right)\left(n-2\right)\cdots\left(n-r+1\right)}{r!}x^r+\cdots\infty\)

**Binomial Coefficient **

The binomial coefficients are the numbers linked with the variables x, y, in the expansion of \( (x+y)^{n}\). The binomial coefficients are represented as \(^nC_0,^nC_1,^nC_2\cdots\) The binomial coefficients can also be obtained by the pascal triangle or by applying the combinations formula.

Learn the various concepts of the Permutations and Combinations here.

The expansion of a binomial raised to some given power is said to be binomial theorem expansion. Different terms related to Binomial expansion involve:

- General term
- Middle term
- Independent term
- To determine a particular term
- Numerically greatest term
- The ratio of consecutive terms also recognised as the coefficients

**General Term of Binomial Expansion**

In the binomial theorem expansion of \((a+b)^{n}\):

\(\left(a+b\right)^n=^nC_0a^nb^0+^nC_1a^{n-1}b^1+^nC_2a^{n-2}b^2+\cdots\cdots+^nC_na^0b^n\)

The general term of binomial expansion is given by the formula: \(T_{r+1}=^nC_r⋅a^{n-r}⋅b^r\)

**Middle Term of Binomial Expansion**

The middle term in the binomial expansion relies on the value of n. The middle term and the number of middle terms depend on the value of n: that is either if n is even or odd.

In the expansion of \((a+b)^{n}\):

\(\left(a+b\right)^n=^nC_0a^nb^0+^nC_1a^{n-1}b^1+^nC_2a^{n-2}b^2+\cdots\cdots+^nC_na^0b^n\)

The middle term is the \(\left(\frac{n}{2}+1\right)\) if n is even.

In the expansion of \((a+b)^{n}\), if n is odd then there are two middle terms which are given by:

\(\frac{\left(n+1\right)}{2}th\ and\ \ \left(\frac{n+1}{2}+1\right)th\ term.\)

Also, read about Arithmetic progressions with this article.

**Independent Term of Binomial Expansion**

The term Independent of in the expansion of

\( \left[ax^p+\left(\frac{b}{x^q}\right)\right]^n\)

Is \(T_{r+1}=^nC_r⋅a^{n-r}⋅b^r\) where r = \(r=\left(\frac{np}{p+q}\right) (integer)\)

That implies in the expansion of \((a+b)^{n}\), the term which is free from the variables, is known as the independent term.

**Numerically Greatest Term of Binomial Expansion**

The numerically greatest term in the expansion of \((a+b)^{n}\):

\(\left(a+b\right)^n=^nC_0a^nb^0+^nC_1a^{n-1}b^1+^nC_2a^{n-2}b^2+\cdots\cdots+^nC_na^0b^n\)

If\(\frac{\left[(n+1)\left|b\right|\right]}{[\left|b\right|+1]}=P\), is a positive integer then \(P^{th}\) term and \((P+1)^{th}\) terms are numerically the greatest terms in the expansion of \((a+b)^{n}\).

Also, read about Relations and Functions here.

**Identifying a Particular Term**

There are two easy actions to recognise a particular term including \(x^{p}\). First, we require to obtain the general term in the expansion of \( (x+y)^{n}\). which is \(T_{r+1}=^nC_rx^{n-r}y^{r}\). Next, we require to compare this with \( x^{p}\) to get the r-value. Here the r-value is essential to find the particular term in the binomial expansion.

Check out this article on Complex Numbers.

**Some other useful Binomial Expansions**

\(\left(x+y\right)^n+\left(x−y\right)^n=2\left[C_0x^n+C_2x^{n-1}y^2+C_4x^{n-4}y^4+\dots\right]\)

\(\left(x+y\right)^n-\left(x−y\right)^n=2\left[C_1x^{n-1}y+C_3x^{n-3}y^3+C_5x^{n-5}y^5+\dots\right]\)

\(\left(1+x\right)^n=\sum_{r-0}^n\ ^nC_r.x^r=\left[C_0+C_1x+C_2x^2+\dots C_nx^n\right]\)

\(\left(1+x\right)^n+\left(1-x\right)^n=2\left[C_0+C_2x^2+C_4x^4+\dots\right]\)

\(\left(1+x\right)^n−\left(1-x\right)^n=2\left[C_1x+C_3x^3+C_5x^5+\dots\right]\)

Check out this article on Rolle’s Theorem and Lagrange’s mean Value Theorem**.**

The binomial theorem expansion also practices over exponents with negative values. The standard coefficient states of binomial expansion for positive exponents are the equivalent for the expansion with the negative exponents. Some of the binomial formulas for negative exponents are as follows:

\((1+x)^{-1}=1-x+x^2-x^3+x^4-x^5+\cdots\)

\((1-x)^{-1}=1+x+x^2+x^3+x^4+x^5+\cdots\)

\((1+x)^{-2}=1-2x+3x^2-4x^3+\cdots\)

\((1-x)^{-2}=1+2x+3x^2+4x^3+\cdots\)

\((1+x)^{-3}=1-3x+6x^2-10x^3+15x^4+\cdots\)

\((1-x)^{-3}=1+3x+6x^2+10x^3+15x^4+\cdots\)

**Properties of Binomial Theorem Expansion**

There are various properties of binomial theorems which are helpful in mathematical computations. Some of the important binomial coefficient properties are as follows:

- Every binomial expansion possesses an additional one term more than the number shown as the power on the binomial.
- Exponents of every term in the expansion, if combined, give the sum equivalent to the power on the binomial.
- The powers of the first element in the binomial are reduced by one with each succeeding term in the expansion; also the powers on the second term increment by one.
- It should be noted the coefficients develop a symmetrical pattern.

Learn more about Sequences and Series here.

The formula for combinations is applied to determine the value of the binomial coefficients in the expansions using the binomial theorem. The formula to find the combinations of r things taken from n distinct objects is \(^nC_r=\frac{n!}{r!\left(n−r\right)!}\)

- Here the coefficients have the following properties.

\(\^nC_0+^nC_1+\dots.+^nC_n=2^n\)

\(^nC_0+^nC_2+^nC_4+\dots.=2^{n-1}\)

\(^nC_1+^nC_3+^nC_5+\dots.=2^{n-1}\)

\(^nC_0-^nC_1+^nC_2-\dots.+\left(-1\right)^{nn}C_n=0\)

**Some other properties of the binomial theorem are as follows:**

- This binomial expansion formula provides the expansion of \( (x+y)^{n}\) where ‘n’ denotes a natural number. The expansion of \( (x+y)^{n}\) holds (n + 1) terms.
- This binomial expansion formula provides the expansion of \(\left(1+x\right)^n\) where ‘n’ denotes a rational number. This expansion possesses an infinite number of terms.
- If we examine just the coefficients, they are symmetric around the middle term. i.e., the first coefficient is identical to the last one similarly the second coefficient is equivalent to the one that is second from the last and so on.

Learn more about statistics here.

**Pascal’s Triangle**

The values of the binomial coefficients show a particular trend which can be recognised in the form of Pascal’s triangle. Pascal’s triangle is an organisation of binomial coefficients in a triangular pattern. The numbers in Pascal’s triangle hold all the border elements as one and the left numbers within the triangle are located in such a way that each number is the summation of two numbers just above the number. Check out the below figure for more information.

## Binomial Theorem Examples

Below are some of the binomial theorem questions to understand the expansion more clearly:

**Solved Example 1.** What is the value of \(\left(1+5\right)^3\) using binomial expansion?

**Solution:**

The binomial expansion formula is,

\(\left(x+y\right)^n=x^n+nx^{n-1}y+\frac{n\left(n-1\right)}{2!}x^{n-2}y^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^{n-3}y^3+\cdots+y^n\)

From the given equation;

x = 1 ; y = 5 ; n = 3

\(\left(x+y\right)^n=\left(1+5\right)^3\)

\(=\left(1\right)^3+3\left(1\right)^{3-1}\left(5\right)^1+\frac{3\left(3-1\right)}{2!}\left(1\right)^{3-2}\left(5\right)^2+\frac{3\left(3-1\right)\left(3-2\right)}{3!}\left(1\right)^{3-3}\left(5\right)^3\)

\(=1+3\times5+\frac{3\times2}{2!}\times25+\frac{3\times2\times1}{3!}\times125\)

\(=1+15+75+125\)

\(=216\)

Learn more about Limit and Continuity here.

**Solved Example 2. **Find the last digit of \(\left(1021\right)^{3921}+\left(3081\right)^{3921}\).

Given:

\(\left(1021\right)^{3921}+\left(3081\right)^{3921}\)

**Concept used:**

The unit digit of the number having 1 in the unit place always gives the unit digit 1 for every value in the exponential power.

Calculation:

\(\left(1021\right)^{3921}=\left(1020+1\right)^{3921}\text{ and }\left(3081\right)^{3921}=\left(3080+1\right)^{3921}\)

Applying binomial expansion, the last digits are 1 and 1 respectively.

Adding these gives 1+1 = 2 i.e. the last digit is 2.

Learn more about Differential Calculus with this article.

**Solved Example 3.** What is the coefficient of the middle term in the binomial expansion of\(\left(2+3x\right)^4\)?

**Concept:**

General term: General term in the expansion of \( (x+y)^{n}\) is given by the formula:

\(T_{r+1}=^nC_rx^{n-r}y^{r}\)

Middle terms: The middle term is the expansion of \( (x+y)^{n}\)depends upon the value of n.

If n is even, then the total number of terms in the expansion of \( (x+y)^{n}\) is n+1. So there is only one middle term i.e. \(\left(\frac{n}{2}+1\right)\)th term is the middle term.

\(T_{\left(\frac{n}{2}+1\right)}=^nC_{\frac{n}{2}}\times x^{\frac{n}{2}}\times y^{\frac{n}{2}}\)

If n is odd, then the total number of terms in the expansion of \( (x+y)^{n}\) is n+1. So there are two middle terms i.e.

\(\frac{\left(n+1\right)}{2}th\ and\ \ \left(\frac{n+1}{2}+1\right)th\ term.\) are two middle terms.

**Calculation:**

Here, we have to find the coefficient of the middle term in the binomial expansion of \(\left(2+3x\right)^4\).

Here n = 4 (n is an even number)

∴ Middle term =\(\left(\frac{n}{2}+1\right)=\left(\frac{4}{2}+1\right)=3^{\text{rd}} \text{ term }\)

\(T_{r+1}=^nC_rx^{n-r}y^r\Rightarrow T_3=T_{\left(2+1\right)}=^4C_2\left(2\right)^{4-2}\left(3x\right)^2\)

\(T_3=^4C_2\left(2\right)^{4-2}\left(3x\right)^2=6\times4\times9x^2\)

\(T_3=6\times4\times9x^2=216x^2\)

∴ Coefficient of the middle term = 216

Check out this article on Logarithmic functions.

**Solved Example 4.** What will be the first negative term in the expansion of \(\left(1+x\right)^{\frac{3}{2}}\) ?

**Concept:**

General term in the expansion of \((a+b)^{n}\) is given by, \(T_{r+1}=^nC_r⋅a^{n-r}⋅b^r\), where r is never fractional.

\(^nC_r=\frac{\left\{n\times\left(n−1\right)\times…\times\left(n−r+1\right)\right\}}{r!}\)

Calculation:

Given binomial expansion: \(\left(1+x\right)^{\frac{3}{2}}\)

\(T_{r+1}=^{\frac{3}{2}}C_r⋅\left(1\right)^{n-r}⋅\left(x\right)^r\)

\(=\frac{\frac{3}{2}\times\left(\frac{3}{2}−1\right)\times…\times\left(\frac{3}{2}−r+1\right)}{r!}\times\left(x\right)^r\)

Now, for this term to become negative,

\(\left(\frac{3}{2}−r+1\right)<0\)

\(\Rightarrow r>\frac{5}{2}\)

\(\Rightarrow r=2.5\)

⇒r = 3 (∵ r is never fractional)

⇒ r + 1 = 3 + 1 = 4

\(T_{3+1}=\frac{\frac{3}{2}\times\left(\frac{3}{2}−1\right)\times\left(\frac{3}{2}−2\right)}{3!}\times\left(x\right)^3\)

\(=\frac{\frac{3}{2}\times\frac{1}{2}\times\left(-\frac{1}{2}\right)}{3\times2}\times\left(x\right)^3\)

\(=-\frac{1}{16}x^3\)

Check out this article on Number Systems.

**Applications of Binomial Expansion**

The binomial theorem has an extensive range of applications in mathematics for example obtaining the remainder, locating digits of a number, etc. The most popular binomial expansion formula practical applications are as follows:

- Finding Remainder using Binomial Theorem
- Finding Digits of a Number
- Relation Between two Numbers
- Divisibility Test

The binomial theorem expansion is a quick method of opening a binomial expression raised to any power. The binomial equation, in particular, is quite an essential topic in algebra and has notable application in matrices, permutations and combinations, probability, and mathematical induction.

The binomial theorem is extensively applicable in statistical and probability estimations. It is extremely useful as the nation’s economy critically relies on statistical and probability analysis.

Also, read about Arithmetic Mean with this article.

In high-level mathematics and computation, using binomial theorem we can identify the roots of equations in higher powers. In addition to this, it is further applied in determining many essential equations in mathematics and physics. Moreover, binomial theorem rules are used in different mathematical and scientific calculations that are mostly linked to multiple topics including:

- Weather forecasting assistance.
- Architectural outlining and services for evaluating the cost in engineering projects.
- Kinematic and gravitational time expansion

- Kinetic energy
- Electric quadrupole pole
- Defining the relativity factor gamma

Isaac Newton holds pride in developing the general binomial expansion formula. The binomial theorem can further be represented as a never-ending equilateral triangle of algebraic expressions named Pascal’s triangle.

Check more topics of Mathematics here.

**Binomial Theorem Key Takeaways**

The following key points would be helpful in a better understanding of the binomial expansion method.

- The number of terms in the binomial expansion of \( (x+y)^{n}\) is equal to n + 1.
- In the expansion of \( (x+y)^{n}\), the summation of the powers of x and y in every term is equal to n.
- The estimation of the binomial coefficients from both sides of the expansion is equal.
- Applying summation notation, the binomial theorem can be displayed as:

- \(\left(x+y\right)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+^nC_2x^{n-2}y^2+\cdots\cdots+^nC_nx^0y^n\)

- \((x+y)^n=\sum_{k=0}^n\left(^nC_k\right)x^{n−k}y^k=\sum_{k=0}^n\left(^nC_k\right)x^ky^{n−k}.\)
- A
**binomial**is a polynomial consisting of two terms, or monomials, distributed by an addition or subtraction representation. **A binomial coefficient**of any of the terms in the extension of the binomial power \( (x+y)^{n}\)**.****Factorial**: The outcome of multiplying a provided number of consecutive integers from 1 to the assigned number. In equations, it is symbolized by an exclamation mark (!). For example, 6!=1⋅2⋅3⋅4⋅5.6=720.

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**Binomial Expansion FAQs**

**Q.1 What is the binomial theorem?**

**Ans.1**The binomial theorem states the principle for extending the algebraic expression \( (x+y)^{n}\) and expresses it as a summation of the terms including the individual exponents of variables x and y.

**Q.2 What is the need for a binomial theorem?**

**Ans.2**Questions with larger raise to power are lengthy and difficult to calculate, in such cases binomial expression is very helpful as it can be implemented for expanding an expression that has been raised to any finite power/large.

**Q.3 What is the formula for a binomial theorem general term?**

**Ans.3**The general term of binomial expansion is given by the formula: \(T_{r+1}=^nC_r⋅a^{n-r}⋅b^r\)

**Q.4 What is binomial distribution?**

**Ans.4**The binomial distribution is a probability distribution that compiles the possibility that a value will take one of two independent values under a provided set of parameters/assumptions. The binomial distribution is generally employed to discrete distribution in statistics.

**Q.5 Who invented the binomial expansion?**

**Ans.5**Isaac Newton is commonly credited with the generalized binomial theorem, efficient for any rational exponent.

**Q.6 What is the use of combination in the binomial expansion?**

**Ans.6**The formula for combinations is applied to determine the value of the binomial coefficients in the expansions using the binomial theorem. The combinations in this instance are the various ways of choosing variables from the available n variables.

**Q.7 What is the difference between binomial and normal distribution?**

**Ans.7**The normal distribution represents continuous data that has an asymmetric distribution whereas Binomial distribution represents the distribution of binary data from a finite sample. Hence it provides the probability of getting r events out of n trials.

“

**Q.8 What are the different terms related to Binomial expansion?**

**Ans.8**The different terms related to Binomial expansion are:

- General term
- Middle term
- Independent term
- To determine a particular term
- Numerically greatest term