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Binomial Expansion Formula is used to expand binomials with any finite power that cannot be expanded using algebraic identities. It is an algebraic formula that describes the algebraic expansion of powers of a binomial.
- Binomial is an algebraic expression with only two terms such as a + b and x - y.
- Algebraic Identities are used to find the expansion when a binomial is raised to exponents 2 and 3.
- Binomial Expansion Formula is used when the exponents of the binomial are large numbers.
- It is also referred to as the ‘Binomial Theorem’.
Binomial Expansion Formula is given as
| \(\large (x+y)^{n} = x^{n} + nx^{n-1}y + \frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}\) |
Read More: NCERT Solutions For Class 11 Maths Binomial Theorem
Key Terms: Binomial Expansion Formula, Binomial Theorem, Binomials, Algebraic Expressions, Exponents, Pascal's Triangle
What is Binomial Expansion?
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Binomial Expansion is a method used to expand the binomials with exponent powers in algebraic expressions. A binomial is an algebraic expression that has exactly two terms.
- It is used to find the expanded value of the algebraic expression of the form (x + y)n.
- Binomial Exapnaison is also known as Binomial Theorem.
- It is easy to find the values with the power 2 or 3, however, finding the expanded form of (x + y)17 needs too much calculation.
- Thus, Binomial Theorem is used in cases with large exponent values.
- The value of the exponent of binomial expansion can be a negative number or a fraction.
Binomial Theorem and Pascal Triangle
Binomial Expansion Formula
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Binomial Expansion Formula is used to find the powers of the binomials which cannot be expanded using the algebraic identities. It involves binomial coefficients which are of the form \(\left(\begin{array}{l}n \\k\end{array}\right)\)or \(n_{ C_{k}}\). It is calculated using the formula,
| \(^n{C_{k}}= {n! \over (n - k)! k!}\) |
Binomial Expansion Formula of Natural Powers
Binomial Expansion Formula is used to give the expansion of (x + y)n where 'n' is a natural number. The expansion of the binomial (x + y)n has (n + 1) terms.
| (x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn |
Here, the nCk Formula is used to calculate the binomial coefficients which is as follows:
| \(^n{C_{k}}= {n! \over (n - k)! k!}\) |
Applying the given formula, the above binomial expansion formula can also be written as,
| (x + y)n = xn + n xn - 1 y1 + [n(n - 1)/2!] xn-2 y2 + [n(n - 1)(n - 2)/3!] xn - 3 y3 +... + n x yn - 1 + yn |
Important Note: If only coefficients are observed, they are symmetric about the middle term. i.e. the first coefficient is the same as the last one, the second coefficient is as same as the one that is second from the last.
Binomial Expansion Formula of Rational Powers
Binomial Expansion Formula is also used to give the expansion of (1 + x)n where 'n' is a rational number. This type of expansion has an infinite number of terms.
| (1 + x)n = 1 + n x + [n(n - 1)/2!] x2 + [n(n - 1)(n - 2)/3!] x3 +... |
Important Note: In order to use the above formula, the value of |x| should be less than 1.
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Solved Examples on Binomial Expansion Formula
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Given below are a few solved examples on Binomial Expansion Formula:
Example 1: Find the expansion of (x + y)6.
Solution: The given binomial is (x + y)6.
Using Binomial Expansion Formula, we get
(x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn
Thus,
(x + y)6 = 6C0x6 + 6C1x5 y + 6C2x4y2 + 6C3x3y3 + 6C4x2y4 + 6C5xy5 + 6C6y6
= (6!/[(6-0)!0!]) x6 + (6!/[(6-1)!1!]) x5 y + (6!/[(6-2)!2!]) x4y2 + (6!/[(6-3)!3!])x3y3 + (6!/[(6-4)!4!])x2y4 + (6!/[(6-5)!5!])xy5 + (6!/[(6-6)!6!])y6
= x6 + 6x5 y + 15x4 y2 + 20x3 y3 + 15x2 y4 + 6x y5 + y6
Thus, the expansion of (x + y)6 is x6 + 6x5 y + 15x4 y2 + 20x3 y3 + 15x2 y4 + 6x y5 + y6.
Read More: Important Questions for Class 11 Maths Binomial Theorem
Example 2: Determine the expansion of (a + b)3.
Solution: We need to find the expansion of (a + b)3
Using Binomial Expansion Formula, we get
(x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn
Thus,
(a + b)3 = 3C0a3 + 3C1a(3 - 1) b + 3C2a(3 - 2) b2 + 3C3a(3 - 3) b3
= ( 3! / [(3-0)!0!] ) a3 + ( 3! / [(3-1)!1!] ) a(3 - 1) b + ( 3! / [(3-2)!2!] ) a(3 - 2) b2 + ( 3! / [(3-3)!3!] ) a(3 - 3) b3
= (1) a3 + (3) a2 b + (3) a1 b2 + (1) a0b3
= a3 + 3a2 b + 3ab2 + b3
Thus, the expansion of (a + b)3 is a3 + 3a2 b + 3ab2 + b3.
Properties of Binomial Theorem
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Some important properties of the Binomial Theorem are as follows:
- In the binomial expansion of (x+y)n, the number of terms is (n+1).
- xn and yn are the first and the last terms respectively.
- The total of the exponents in each term is n.
- nC0, nC1, nC2, … nCn are called binomial coefficients and also denoted by C0, C1, C2. . . . . Cn.
- The binomial coefficients at the start and at the end are equivalent. i.e. nC0 = nCn, nC1 = nCn-1, nC2 = nCn – 2.
- The exponent of x reduces by 1 from term to term while progressing from the first to the last.
- When the exponent of y increases by one, the exponent of x increases by one.
- The binomial coefficients in the expansion are arranged in an array known as Pascal's Triangle.
Important Terms of Binomial Expansion
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Binomial Theorem is used for the expansion of a binomial raised to some power. It is commonly referred to as Binomial Expansion. Given below are the terms related to binomial expansion using the binomial theorem:
- General Term
- Middle Term
- Identifying a Particular Term
- Numerically Greatest Term
- Independent Term
- Ratio of Consecutive Terms also known as Coefficients
Binomial Expansion of Negative Exponent
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Binomial Expansion is also applicable for exponents with negative values. The standard coefficient values of binomial expansion are the same for positive and negative exponents. The algebraic relationship between the terms varies in the binomial expansion of negative exponents while the terms and the coefficient values remain the same.
- (1 + x)-1 = 1 - x + x2 - x3 + x4 - x5 + .......
- (1 - x)-1 = 1 + x + x2 + x3 + x4 + x5 + .......
- (1 + x)-2 = 1 - 2x + 3x2 - 4x3 + ........
- (1 - x)-2 = 1 + 2x + 3x2 + 4x3 + ........
- (1 + x)-3 = 1 - 3x + 6x2 - 10x3 + 15x4 + ......
- (1 - x)-3 = 1 + 3x + 6x2 + 10x3 + 15x4 + ......
Binomial Theorem and Pascal’s Triangle
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Pascal's Triangle is a triangular pattern of numbers designed by Blaise Pascal. Binomial Expansion of terms can be addressed using Pascal's triangle. Consider a binomial (a + b) which is raised to the force 'n' and let 'n' be an entire number. For allotting the values of to 'n' as {0, 1, 2 … ..}, the binomial expansions of (a+b)n for different values of ‘n’ are shown below:
| Binomial Expansion | |
|---|---|
| (a+b)º | 1 |
| (a+b)¹ | a+b |
| (a+b)² | a² + 2ab + b² |
| (a+b)³ | a³ + 3a²b + 3ab² + b³ |
| (a+b)4 | a4 + 4a³b + 6a²b² + 4ab² + b4 |
| (a+b)5 | a5 + 5a4b + 10a³b² + 10a²b³ + 5ab4 + b5 |
With this representation, the following observations can be made:
- Every expansion has one term more than the chosen value of ‘n’.
- In each term of the expansion, the sum of the power is equivalent to the initial value of chosen 'n'.
- The powers of 'a' start with the chosen value of 'n' and reduces to zero across the terms in expansion while the powers of 'b' start with zero and accomplish the maximum value of ‘n'.
- Coefficients start with 1, increase till halfway and then reduce by the same amounts to end with one.
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Things to Remember
- Binomial Expansion is a method used for the expansion of binomial expressions with large exponential power.
- It is also known as Binomial Theorem and helps in the expansion of a binomial raised to a certain power.
- Binomial Expansion Formula is given as \(\large (x+y)^{n} = x^{n} + nx^{n-1}y + \frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}\).
- The exponent of a binomial expansion can be positive, negative, or a fraction.
- The number of terms in the binomial expansion of (x + y)n is n + 1.
- Pascal's Triangle is a triangular arrangement of numbers that gives the coefficients in binomial expansion.
Previous Years’ Questions
- The coefficient of xn in the binomial expansion of…
- In the binomial expansion of (a - b)n, n≥5 the sum of the 5th…
- The numerically greatest term in the binomial expansion of… (AP EAMCET - 2018)
- In the binomial expansion of (1+x)15 the coefficients of… (KCET - 2010)
- The total number of terms in the expansion of ( 1 + x )2n… (AMUEEE - 2012)
- Sum of last 30 coefficients in the binomial expansion of… (KEAM - 2018)
- Let tn denote the nth term in a binomial expansion. If… (KEAM)
- The coefficient of xn in the expansion of (1+ x)(1- x)n is… (AIEEE - 2004)
- The number of irrational terms in the expansion of… (WBJEE - 2019)
- The coefficient of x50 in the binomial expansion of… (JEE Main - 2014)
Sample Questions
Ques. What is Binomial Expansion? (3 Marks)
Ans. Binomial Expansion is defined as a method in Algebra that is used to expand the binomials with powers in algebraic expressions.
For two terms x and y, the binomial expansion to the power of n is given as
(x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn
Here, the number of terms in the binomial expansion with an exponent of n is n + 1.
Ques. Determine the expansion of (3x + y)1/2 up to the first three terms where \(\left|\dfrac y {3x}\right|<1\). (3 Marks)
Ans. (3x + y)1/2 = 3x (1 + y/(3x))1/2
On Comparing (1 + y/(3x))1/2 with (1 + x)n, we get x =y/(3x) and n = 1/2.
Using Binomial Expansion Formula, the expansion of (1 + y/(3x))1/2 up to the first three terms is,
1 + n x + [n(n - 1)/2!] x2 = 1 + (1/2) (y / (3x)) + [(1/2) ((1/2) - 1)/2!] (y / (3x))2 = 1 + y / (6x) - y2 / (72x2)
3x [1 + y / (6x) - y2 / (72x2) ] = 3x + y / 2 - y2 / (24x)
Therefore, the expansion of 3x (1 + y/(3x))1/2 up to the first three terms is 3x + y / 2 - y2 / (24x).
Ques. What are Binomials? (2 Marks)
Ans. Binomials are polynomials with only two terms. One can understand a binomial as bi which generally means 2 and thus binomial will have two terms. Some examples of binomials are as follows:
- 3x + 2
- 2x + 4y
- x + y
Ques. How to Identify a Binomial Random Variable? (3 Marks)
Ans. One can distinguish a random variable as being Binomial Random Variable if the following four requirements are met:
- There are a fixed number of trials (n).
- Each trial has two possible results either success or failure.
- The likelihood of success (call it p) is the same for each trial.
- The trials are independent of one another.
Ques. Where is Binomial Theorem used? (3 Marks)
Ans. The important uses of the Binomial Theorem are as follows:
- Binomial Theorem is used majorly in Statistical and Probability Analyses.
- Binomial Theorem is used in attaining roots of equations with higher powers.
- It is also used in probability for binomial expansion.
Ques. What is Binomial Expansion Formula? (3 Marks)
Ans. Binomial Expansion Formula is used to find the expanded form of a binomial raised to a certain power. Binomial Expansion Formulas are as follows:
(x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn
Where 'n' is a natural number and nCk = n! / [(n - k)! k!].
(1 + x)n = 1 + n x + [n(n - 1)/2!] x2 + [n(n - 1)(n - 2)/3!] x3 +...
When 'n' is a rational number and |x| < 1.
Ques. What are n and r in Binomial Expansion Formula? (1 Mark)
Ans. In the Binomial Expansion Formula for (x+a)n, the combinatorics formula is used that is denoted as nCr, where n is the exponent in the expansion and r is the term number that ranges from 0 to n.
Ques. How to determine the terms in Binomial Theorem?? (2 Marks)
Ans. In a Binomial Expansion with an exponent of n, the number of terms is equal to n + 1. Also, in order to find a particular term in the expansion of (x + y)n, the general term formula is used. The general term of the binomial expansion is given as Tr+1 = nCr xn-ryr. The values of the coefficients are found from the Pascals Triangle or Combinations Formula. The sum of the exponents of both terms in the general term is equal to n.
Ques. How is Binomial Expansion Formula derived? (3 Marks)
Ans. Binomial Expansion Formula is given as
(x + y)n = nC0xn y0 + nC1xn - 1 y1 + nC2xn-2 y2 + nC3xn - 3 y3 + ... + nCn−1x yn - 1 + nCnx0yn
It is derived using mathematical induction using the given steps:
- First, prove the formula for n = 1.
- Now, assume that the formula is true for n = k.
- Lastly, prove the formula for n = k.
Ques. What are the uses of Binomial Expansion Formula? (1 Mark)
Ans. Binomial Expansion Formula is mainly used to find the power of a binomial without actually multiplying the binomial by itself many times. It is used in various concepts of math such as algebra, calculus, combinatorics, etc.
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