Binomial expansion formula is a formula that is used to solve binomial expressions. A binomial is an algebraic expression with two terms. For example, x + y, x - a, etc are binomials.
In this article, we have covered the Binomial Expansion definition, formulas, and others in detail.
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Binomial Expansion
An algebraic expression containing two terms is called a binomial expression. Example: (x + y), (2x - 3y), (x + (3/x)). The general form of the binomial expression is (x + a) and the expansion of (x + a)n, n ∈ N is called the binomial expansion. The binomial expansion provides the expansion for the powers of binomial expression.
What Are Binomial Expansion Formula?
Binomial expansion formulas are formulas that are used to solve algebraic expressions which are not easily solved using algebraic identities. Binomial Expansion Formulas are categorized into two categories that are:
- Binomial Expansion Formula of Natural Powers
- Binomial Expansion Formula of Rational Powers
Binomial Expansion Formula of Natural Powers
Binomial expansion formula for the expansion of (x + y)n where 'n' is a natural number is added below:
(x + a)n = nC0xna0 + nC1xn-1a1 + nC2xn-2a2 + .........+ nCr xn-rar + ........ + nCn-1x1an-1 + nCnx0an
(x + a)n = nCr xn-rar
Proof:
Proof of binomial expansion using the principle of mathematical induction on n.
Let X(n) be : (x + a)n = nC0xna0 + nC1xn-1a1 + nC2xn-2a2 + .........+ nCr xn-rar + ........ + nCn-1x1an-1 +nCnx0an
Step I:
To prove: X(1) : (x + a)1 =1C0x1a0 + 1C1x0a1
We know that : (x + a)1 = x + a = 1C0x1a0 + 1C1x0a1
therefore, X(1) is true
Step II:
Let X(m) be true. Then,
(x + a)m = mC0xma0 + mC1xm-1a1 + mC2xm-2a2 + .........+ mCm-1x1am-1 +mCmx0am ------------(1)
To prove: X(m+1) is true. i.e.
(x + a)m+1 = m+1C0xm+1a0 + m+1C1xma1 + m+1C2xm-1a2 + .........+ m+1Cmx1am +m+1Cm+1x0am+1
Proof: (x + a)m+1 = (x + a)(x + a)m
= (x + a)[mC0xma0 + mC1xm-1a1 + mC2xm-2a2 + .........+mCrxm-rar+ mCm-1x1am-1 +mCmx0am]
= mC0xm+1a0 + (mC1 + mC0)xma1 + (mC2 + mC1)xm-1a2 + ... +(mCr + mCr-1)xm-r+1ar + ... + (mCm-1 + mCm)x1am + mCmam+1
[Since, mCr-1 + mCr = m+1Cr , r = 1, 2, 3....., m]
= m+1C0xm+1a0 + m+1C1xma1 + m+1C2xm-1a2 + .........+ m+1Cmx1am + m+1Cm+1x0am+1
X(m + 1) is true.
X(m) is true ⇒ X(m + 1) is true
Binomial Expansion Formula of Rational Powers
Binomial expansion formula for the expansion of (1 + x)n where 'n' is a rational number is added below:
(1 + x)n = 1 + n x + [n(n - 1)/2!] x2 + [n(n - 1)(n - 2)/3!] x3 + ...
Binomial Expansion Formula Characterstics
Various characteristics of binomial expansion formulas are:
- In Binomial expansion, r can have values from 0 to n, the total number of terms in the expansion is (n+1).
- Sum of indices of x and a in each term is n.
- Since, nCr = nCn-r , for r = 0,1,2......,n . Hence, the coefficients of terms equidistant from the starting and end are equal. So such coefficients are known as binomial coefficients.
- (x-a)n =
\sum_{r=0}^{n} (-1)r nCrxn-rar In the expansion of (x-a)n we have alternate positive and negative terms and sign of last term depends on the value of n (odd or even). - Coefficient of (r+1)th term or xr in the expansion of (1 + x)n is nCr.
- (x + a)n + (x - a)n = 2[nC0xna0 + nC2xn-2a2 + .......] and (x + a)n - (x - a)n = 2[nC1xn-1a1 + nC3xn-3a3 + .......]
- In the binomial expansion of (x + a)n, the general term is given by Tr+1 = nCrxn-rar
- In the binomial expansion of (x - a)n, the general term is given by Tr+1 = (-1)r nCrxn-rar
- Binomial expansion of (x + a)n contains (n + 1) terms. Therefore, if n is even, then ((n/2) + 1)th term is the middle term and if n is odd, then ((n + 1)/2)th and ((n + 3)/2)th terms are the two middle terms.
Different values of n have a different number of terms:
| n | (x + a)n + (x - a)n | (x + a)n - (x - a)n |
|---|---|---|
| odd | (n+1)/2 | (n+1)/2 |
| even | (n/2)+1 | (n/2) |
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Examples Using Binomial Expansion Formulas
Example 1: Find the number of terms in the expansions of the following :
(i) (9x - y)9
(ii) (1 +7x)9 + (1 - 7x)9
(iii) (1 + 2x + x2)20
Solution:
(i)
In the expansion of (x + a)n the number of terms is (n+1)
Hence, in the expansion of (9x - y)9 the number of terms is 10
(ii)
In the expansion of (x + a)n + (x - a)n the number of terms is (n+1)/2 if n is odd.
So number of terms in the expansion of (1 + 7x)9 + (1 - 7x)9
= (10/2) = 5
(iii)
(1 + 2x + x2)20
= [(1 + x)2]20
= (1 + x)40
Hence, number of terms = 41
Example 2: Expand (3x + 8)4
Solution:
According to binomial expansion :
(3x + 8)4 = 4C0 (3x)4 (8)0 + 4C1 (3x)3 (8)1 + 4C2 (3x)2 (8)2 + 4C3 (3x)1 (8)3 + 4C4 (3x)0 (8)4
= (3x)4 + 4.(3x)3.8 + 6.(3x)2.64 +4.(3x).512 + 4096
= 12x4 + 864 x3 + 3456 x2 + 6144 x + 4096
Example 3: Expand (2x - 1)5
Solution:
According to binomial expansion :
(2x - 1)5 = (2x + (-1))5 = 5C0 (2x)5 (-1)0 + 5C1 (2x)4 (-1)1 + 5C2 (2x)3 (-1)2 + 5C3 (2x)2 (-1)3 + 5C4 (2x)1 (-1)4 + 5C5 (2x)0(-1)5
= 32x5 - 5.16x4 + 10.8x3 - 10.4x2 + 10x - 1
= 32x5 - 80x4 + 80x3 - 40x2 + 10x - 1
Example 4: Expand (1 + x + x2)3
Solution:
Let, y = x + x2
(1 + x + x2)3 = (1 + y)3 = 3C0 (1)3 (y)0 + 3C1 (1)2 (y)1 + 3C2 (1)1 (y)2 + 3C3 (1)0 (y)3
= 1 + 3y + 3y2 + y3
= 1 + 3(x + x2) + 3(x + x2)2 + (x + x2)3 = 1 + 3x + 3x2 + 3(x2 + x4 + 2x3) + (x3 + x6 + 3x4 + 3x5)
= 1 + 3x + 3x2 + 3x2 + 3x4 + 6x3 + x3 + x6 + 3x4 + 3x5
= 1 + 3x + 6x2 + 7x3 + 6x4 + 3x5 + x6
Example 5: Find (a + b)4 - (a - b)4. Hence, evaluate (√3 + √2)4 - (√3 - √2)4.
Solution:
(a + b)4 - (a - b)4 = 2.[4C1a3b1 + 4C3a1b3] = 2.[4a3b1 + 4a1b3] = 8a3b1 + 8a1b3
Put a = √3 and b = √2
(√3 + √2)4 - (√3 - √2)4 = 8.(√3)3(√2) + 8.(√3)(√2)3 = 24√6 + 16√6 = 40√6
Example 6: Find the 10th term in the binomial expansion of (4x2 + 1/x)11.
Solution:
In the binomial expansion of (x + a)n , (r+1)th term is given by Tr+1 = nCrxn-rar
In the expansion of (4x2 + 1/x)11 , [n = 11, r = 9, x = 4x2, a = 1/x]
T10 = T9+1 = 11C9 (4x2)11-9 (1/x)9 = 55.(16x4).(1/x9) = 880/x5
Example 7: Find the middle term in the expansion of [(4/3)x2 - (3/4x)]20.
Solution:
Here, n = 20 (even)
[(20/2) + 1]th term i.e. 11th term is the middle term.
Hence, the middle term = T11 = T10+1 = 20C10.[(4x2/3)]20-10. [-(3/4x)]10 = 20C10.x10
Summary
Binomial Expansion is a mathematical technique used to expand expressions that are raised to a power, where the expression is a binomial (an algebraic expression with two terms). The binomial expansion of
Practice Problems on Binomial Expansion Formulas
Question 1: Find the number of terms in the expansion of
Question 2: Expand
Question 3: Expand
Question 4: Find the 8th term in the expansion of
Question 5: Determine the middle term in the expansion of