Binomial Theorem and Important Terms of Binomial Expansion

Binomial Theorem

A. Binomial Theorem

The Binomial Theorem gives a formula to expand any positive integral power of a binomial expression into a sum of terms involving binomial coefficients.

The expansion of (x + y)ⁿ is

(x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 y + ⁿC₂ x^n-2 y² + ··· + ⁿC_r x^n-r y^r + ··· + ⁿC_n yⁿ

The theorem can be proved by mathematical induction on n. The coefficients ⁿC_r are the binomial coefficients, defined by

ⁿC_r = n! / (r! (n - r)!), for 0 ≤ r ≤ n.

Observations :

• The number of terms in the expansion is (n + 1), i.e. one more than the exponent n.
• The sum of the exponents of x and y in each term is n.
• Binomial coefficients equidistant from the beginning and the end are equal: ⁿC_r = ⁿC_n-r.

Example 1

Ex.1 The value of

Sol.

The numerator is of the form a³ + b³ + 3ab(a + b) = (a + b)³, where a = 18 and b = 7.

Numerator = (18 + 7)³ = 25³.

The denominator can be written as

= (3 + 2)⁶ = 5⁶ = 25³.

Therefore the value is 1.

B. IMPORTANT TERMS IN THE BINOMIAL EXPANSION

(a) General term

The general term or the (r + 1)^th term in the expansion of (x + y)ⁿ is

T_r+1 = ⁿC_r x^n-r y^r, for r = 0, 1, 2, ..., n.

Example 2

Ex.2 Find : (a) The coefficient of x⁷ in the expansion of

(b) The coefficient of x^-7 in the expansion of

Also, find the relation between a and b, so that these coefficients are equal.

Sol.

(a) Consider the expansion of

The general term is

Equate the exponent of x to 7: 22 - 3r = 7.

3r = 15.

r = 5.

Hence the coefficient of

(b) Consider the expansion of

The general term is

Equate the exponent of x to -7: 11 - 3r = -7.

3r = 18.

r = 6.

Hence the coefficient of

Also given coefficient of

= coefficient of

Therefore ab = 1 (since ¹¹C₅ = ¹¹C₆), which is the required relation between a and b.

Hence the coefficient of x^-7 in

Also given coefficient of x⁷ in

¹¹C₅ a⁶ b^-5 = ¹¹C₆ a⁵ b^-6.

Therefore ab = 1 (because ¹¹C₅ = ¹¹C₆), as required.

Example 3

Ex.3 Find the number of rational terms in the expansion of (9^1/4 + 8^1/6)¹⁰⁰⁰.

Sol.

The general term in the expansion of (9^1/4 + 8^1/6)¹⁰⁰⁰ is

The general term is rational when the exponents of prime factors (3 and 2) are integers.

The exponent of 3 in the term is 1000 - r/2.

The exponent of 2 in the term is r/2.

Both 1000 - r/2 and r/2 must be integers, so r must be even.

The possible values of r are {0, 2, 4, ..., 1000}.

Hence the number of rational terms is 1000/2 + 1 = 501.

(b) Middle term

The middle term(s) in the expansion of (x + y)ⁿ are:

• If n is even, there is one middle term: T_(n+2)/2 = ⁿC_n/2 x^n/2 y^n/2.
• If n is odd, there are two middle terms: T_(n+1)/2 and T_(n+3)/2.

Example 4

Ex.4 Find the middle term in the expansion of

Sol.

The number of terms in the expansion of

is 10 (even). So there are two middle terms.

The two middle terms are T₅ and T₆.

(c) Term independent of x

The term independent of x (constant term) is the term in which the exponent of x is zero. To find it, equate the exponent of x in the general term to zero and solve for r.

Example 5

Ex.5 The term independent of x in

Sol.

The general term in the expansion is

For a constant term we require 3r/2 = 10.

r = 20/3, which is not an integer.

Therefore there is no constant term in this expansion.

(d) Numerically greatest term

To find the numerically greatest term in the expansion of (x + a)ⁿ for a given x, compare successive terms T_r and T_r+1 using the ratio T_r+1 / T_r. If this ratio > 1 then T_r+1 > T_r; if < 1 then T_r > T_r+1. Use this comparison to find the index r for which the term is maximum.

Let T_r and T_r+1 be consecutive terms in the expansion of (1 + a/x)ⁿ (or after factoring out common powers). One derives an inequality for r and locates the greatest term.

If r is a positive integer m and a fraction f such that 0 < f < 1 and r < m + f, then T_m+1 is the greatest term. If r < m (i.e. integer), then T_m and T_m+1 may both be greatest and equal.

Example 6

Ex.6 Find numerically the greatest term in the expansion of (3 - 5x)¹¹ when x = 1/5

Sol.

Since

We compute the ratio of successive terms and find the range of r for which terms increase and then decrease:

Thus the greatest terms occur for r = 2 and r = 3, i.e. T₃ and T₄.

The greatest term when r = 2 is:

The greatest term when r = 3 is:

Both greatest terms have equal numerical values.

C. A USEFUL RESULT FOR (√A + B)ⁿ

Let (√A + B)ⁿ = I + f, where I is an integer part and f is the fractional part with 0 < f < 1.

If n is odd and I and n are positive integers, then (I + f) f = Kⁿ, where A - B² = K > 0 and A - B < 1.

If n is even, then (I + f)(1 - f) = Kⁿ.

Example 7

Ex.7 If (6√6 + 14)²ⁿ⁺¹ = [N] + F and F = N - [N] ; where [*] denotes greatest integer, then NF is equal to

Sol.

Since (6√6 + 14)²ⁿ⁺¹ = [N] + F, let us assume that f = (6√6 - 14)²ⁿ⁺¹, where 0 ≤ f < 1.

[N] + F - f is an even integer.

Now 0 < F < 1 and 0 < f < 1 so -1 < F - f < 1 and F - f is an integer; hence F - f = 0.

Thus

NF = (6√6 + 14)²ⁿ⁺¹ (6√6 - 14)²ⁿ⁺¹ = 20²ⁿ⁺¹.

This completes the evaluation of NF.

Summary

The Binomial Theorem provides a systematic way to expand (x + y)ⁿ and identify specific terms: the general term, middle term(s), constant term, and numerically greatest term. Use binomial coefficients ⁿC_r, check exponents for integrality when powers are fractional, and compare successive terms to find maxima. The examples illustrate extraction of coefficients, counting rational terms, locating middle and constant terms, and special identities involving conjugate expressions.