Binomial Theorem | Full Chapter in ONE SHOT | Chapter 7 | Class 11 Maths 🔥

Summary

The lesson develops the Binomial Theorem from patterns in simple expansions, then applies it to algebraic expansion, approximation, comparisons, and divisibility proofs. The teacher emphasizes that the ideas are straightforward, but calculations can be lengthy, so expansions should be written systematically.

Patterns in binomial expansions

A binomial is an expression with two terms, such as (x+y). In expansions of ((x+y)^n):

• There are (n+1) terms.
• The exponent of the first term decreases from (n) to (0).
• The exponent of the second term increases from (0) to (n).
• Every term has total degree (n).
• The coefficients follow the rows of Pascal’s triangle.

The first few coefficient rows are:

(n) Coefficients 0 (1) 1 (1,1) 2 (1,2,1) 3 (1,3,3,1) 4 (1,4,6,4,1)

Each interior entry in Pascal’s triangle is the sum of the two entries immediately above it.

Binomial coefficients

The coefficients can also be written using combinations. In the expansion of ((x+y)^n), they are (\binom n0,\binom n1,\ldots,\binom nn). Useful facts include:

• (\binom n0=\binom nn=1)
• (\binom n1=n)
• Symmetry: (\binom nr=\binom n{n-r})

These facts can help identify coefficients without calculating every combination from scratch.

The Binomial Theorem

The general expansion is

[ (x+y)^n=\sum_{r=0}^{n}\binom nr x^{\,n-r}y^r. ]

Written out, the first few terms are

[ \binom n0x^n+\binom n1x^{n-1}y+\binom n2x^{n-2}y^2+\cdots+\binom nny^n. ]

A systematic approach to expanding a binomial is to identify the two terms and the exponent (n), write (n+1) terms, and then arrange the coefficients and powers in their respective patterns. Substitute the expressions and simplify only after the structure is in place.

For example,

[ (x+2a)^5=x^5+10x^4a+40x^3a^2+80x^2a^3+80xa^4+32a^5. ]

Expansions involving subtraction

For ((x-y)^n), the powers follow the same pattern, but the signs alternate:

[ (x-y)^n=x^n-\binom n1x^{n-1}y+\binom n2x^{n-2}y^2-\cdots. ]

Odd powers of the negative second term produce negative signs; even powers produce positive signs. For instance,

[ (2x-3y)^4 =16x^4-96x^3y+216x^2y^2-216xy^3+81y^4. ]

Handling more complicated expressions

A three-term expression raised to a power can sometimes be treated as a binomial by grouping two terms together as one term. Expand using the theorem, then simplify the grouped expression.

The theorem can also be applied more than once when a powered expression contains another binomial. Such problems may require substantial simplification, so keeping terms organized is important.

Applications and examples

• Computing large powers: Rewrite a number near a convenient base. For example, (99^5=(100-1)^5), which can be expanded around (100).

• Approximation and comparison: In ((1+0.01)^{1{,}000{,}000}), all terms are positive. The first two terms alone give the lower bound [ 1+1{,}000{,}000(0.01)=10{,}001. ]

• Conjugate expressions: When adding the expansions of ((\sqrt2+1)^6) and ((\sqrt2-1)^6), terms with opposite signs cancel in pairs. The sum is (198).

• Summation notation: The theorem can be written compactly with sigma notation, with the index running from (0) to (n).

Divisibility proofs using expansion

The Binomial Theorem can help reveal common factors:

• To show that (9^{n+1}-8n-9) is divisible by (64), write (9^{n+1}=(1+8)^{n+1}). The constant and linear terms in the expansion are (1) and (8(n+1)). Subtracting (8n+9) removes these terms, while every remaining term contains (8^2=64).
• To show that (a-b) is a factor of (a^n-b^n), rewrite (a=b+(a-b)) and expand ([b+(a-b)]^n). After subtracting (b^n), every remaining term contains (a-b).

The general strategy is to rewrite an expression so the desired factor appears in the binomial, expand, and identify that factor in the resulting terms.

Speakers and sources

• Speaker: One male mathematics instructor; his name is not identified in the subtitles.
• References: Pascal’s triangle and Pascal’s work are discussed, and NCERT-style questions are referenced. No separate guest speaker is identifiable.