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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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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.
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 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 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 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 + ...
Various characteristics of binomial expansion formulas are:
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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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
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 n allows us to express it as a sum of terms involving different powers of x and a with coefficients known as binomial coefficients.
Question 1: Find the number of terms in the expansion of 7
Question 2: Expand 6 and simplify the expression.
Question 3: Expand 8 up to the term involving 2
Question 4: Find the 8th term in the expansion of 9.
Question 5: Determine the middle term in the expansion of 10
