Class 12 Mathematics BINOMIAL THEOREM Notes

Class 12 Mathematics BINOMIAL THEOREM Notes
Class 12 Mathematics Binomial Theorem Notes | Nepal eNotes

Unit 2

Algebra

Class 12 Mathematics

Binomial Theorem

Class 12 Mathematics – Binomial Theorem Notes PDF

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Contents / Quick Navigation

1. Introduction 2. Positive Integral Index 3. Proof of Binomial Theorem 4. General Term 5. Binomial Coefficients 6. Middle Term(s) 7. Independent / Specified Term 8. Binomial Theorem for Any Index 9. Approximation 10. Euler’s Number 11. eˣ, aˣ and log(1+x) 12. Worked Examples 13. Common Mistakes 14. Important Exam Questions 15. One-Minute Revision 16. Diagram Practice

NEB / CDC Focus

This chapter studies the expansion of powers of a binomial. The syllabus focus is the binomial theorem for a positive integral index, its general term and coefficients, the binomial theorem for a general index without proof, applications to approximation, Euler’s number, and standard expansions related to ex, ax and log(1+x).

1. Introduction

Definition

A binomial is an algebraic expression containing two terms, such as a+x, x-y or 1+t.

The binomial theorem provides a systematic way to expand powers such as (a+x)n without repeated multiplication.

PowerExpansion
(a+x)01
(a+x)1a+x
(a+x)2a2+2ax+x2
(a+x)3a3+3a2x+3ax2+x3
Pascal Triangle → Binomial Coefficients 1 11 121 1331 14641 15101051 Each interior coefficient is the sum of the two coefficients directly above it.
Figure 1: Pascal’s triangle displays the coefficients of successive positive-integral binomial expansions.

2. Binomial Theorem for a Positive Integral Index

Binomial Theorem
(a+x)n = nC0an + nC1an-1x + nC2an-2x2 + … + nCran-rxr + … + nCnxn

where n is a non-negative integer.

2.1 Important Observations

  • The expansion contains exactly n+1 terms.
  • The power of a decreases from n to 0.
  • The power of x increases from 0 to n.
  • In every term, the sum of exponents of a and x is n.
  • The coefficients are nC0, nC1, …, nCn.
  • First and last coefficients are both 1.
Pattern in (a+x)ⁿ 1st term ⁿC₀ aⁿ 2nd term ⁿC₁ aⁿ⁻¹x (r+1)th term ⁿCᵣ aⁿ⁻ʳxʳ last term ⁿCₙ xⁿ Power of x increases → Power of a decreases in the same step.
Figure 2: Coefficient and exponent pattern in the binomial theorem.

3. Proof for a Positive Integral Index

The theorem can be understood combinatorially by writing (a+x)n as a product of n identical factors:

(a+x)(a+x)…(a+x).

To obtain a term containing xr, choose x from exactly r of the n factors and choose a from the remaining n-r factors.

The number of ways to choose those r factors is nCr. Hence the contribution is

nCran-rxr.

As r runs from 0 to n, adding all possible terms gives the binomial expansion.

Why the coefficient is ⁿCᵣ (a+x) (a+x) (a+x) (a+x) … Choose x from r of the n factors Number of choices = ⁿCᵣ
Figure 3: Combinatorial reasoning behind the coefficient of the general term.
Exam Important

For a proof question, clearly state why selecting x from r of the n factors gives nCr.

4. General Term

General Term
Tr+1 = nCran-rxr,   r=0,1,2,…,n

The notation is important: when the exponent of x is r, that expression is the (r+1)th term, not the rth term.

Worked Example: Find the 5th term

Find the 5th term in (2x+3)8.

  1. Use Tr+1. For the 5th term, r=4.
  2. T5=8C4(2x)8-434.
  3. =70×16x4×81.
  4. =90,720x4.

Answer: 90,720x4.

4.1 Term from the End

Since the expansion has n+1 terms, the kth term from the end is the (n-k+2)th term from the beginning.

4.2 Ratio of Consecutive Terms

For Tr+1=nCran-rxr,

Tr+2Tr+1 = n-rr+1 · xa

This relation is useful for comparing successive terms or coefficients when needed.

5. Binomial Coefficients

Definition

The numbers nC0, nC1, …, nCn are called binomial coefficients.

Coefficient Formula
nCr = n!r!(n-r)!

5.1 Main Properties

PropertyMeaning
nC0=nCn=1First and last coefficients are 1.
nCr=nCn-rCoefficients are symmetric.
nCr-1+nCr=n+1CrPascal identity.
Σr=0n nCr=2nSet a=x=1 in the binomial theorem.
Σr=0n(-1)rnCr=0, n≥1Set a=1, x=-1.
Symmetry of Binomial Coefficients ⁿC₀ ⁿC₁ … … ⁿCₙ₋₁ ⁿCₙ Equal distances from the two ends give equal coefficients.
Figure 4: The coefficient sequence is symmetric because nCr=nCn-r.

6. Middle Term or Middle Terms

The number of terms in (a+x)n is n+1.

Middle-Term Rules

If n is even, then n+1 is odd, so there is one middle term:

Middle term = T(n/2)+1.

If n is odd, then n+1 is even, so there are two middle terms:

Middle terms = T(n+1)/2 and T(n+3)/2.
Worked Example

Find the middle term of (x+2)8.

Here n=8, so there are 9 terms. The middle term is the 5th term.

T5=8C4x424 =70×16x4=1120x4.

7. Term Independent of x and Specified-Power Terms

For expressions involving positive and negative powers of x, write the general term and set the total power of x equal to the required exponent.

Worked Example: Term independent of x

Find the term independent of x in (x2+1/x)6.

  1. General term: Tr+1=6Cr (x2)6-r(x-1)r.
  2. Power of x is 2(6-r)-r=12-3r.
  3. For an independent term, set 12-3r=0, giving r=4.
  4. Required term: 6C4=15.

Answer: 15.

Important Note

If solving the exponent equation gives a non-integer value of r, or a value outside 0≤r≤n, that requested power does not occur in the expansion.

8. Binomial Theorem for Any Index

For a general real or rational index n, the binomial expansion of (1+x)n is

General Binomial Expansion
(1+x)n =1+nx+ n(n-1)2!x2 + n(n-1)(n-2)3!x3 +…

Convergence condition: typically |x|<1 for the infinite expansion.

Unlike the positive-integer case, this expansion generally continues indefinitely when n is not a non-negative integer.

8.1 Standard Special Forms

ExpressionExpansion for |x|<1
(1+x)-1 1-x+x2-x3+…
(1-x)-1 1+x+x2+x3+…
(1+x)1/2 1+x/2-x2/8+x3/16-…
(1+x)-1/2 1-x/2+3x2/8-5x3/16+…
Remember / Warning

The CDC scope asks for the theorem for any index without proof. Focus on stating it correctly and applying it.

9. Application to Approximation

If |x| is small, higher powers x2,x3,… become much smaller. Therefore, a few leading terms often give a useful approximation.

First-Order Approximation
(1+x)n ≈ 1+nx, when |x| is small.
Second-Order Approximation
(1+x)n ≈ 1+nx+ n(n-1)2x2.
Why small x allows approximation x=0 neighbourhood exact curve few-term approximation Near x = 0, neglected higher powers contribute less.
Figure 5: Truncated binomial expansions approximate the full expression most effectively when |x| is small.
Worked Example: Approximate √1.04

Write √1.04=(1+0.04)1/2.

Using terms up to x2:

(1+x)1/2≈1+x/2-x2/8.

√1.04≈1+0.04/2-(0.04)2/8 =1+0.02-0.0002=1.0198.

Approximate value: 1.0198.

10. Euler’s Number e

Definition

Euler’s number is the mathematical constant e≈2.718281828….

One important defining limit is

e = limn→∞ (1+1/n)n.

A closely related form is ex=limn→∞(1+x/n)n. Applying the binomial theorem to this expression leads to the standard exponential series.

(1 + 1/n)ⁿ approaches e e ≈ 2.71828 As n increases, the sequence approaches the horizontal level e.
Figure 6: Conceptual representation of Euler’s number as a limiting value.

11. Standard Expansions Using the Binomial-Theorem Framework

11.1 Expansion of ex

ex =1+x+ x22! + x33! +…,   -∞<x<∞

11.2 Expansion of ax

Since ax=ex\ln a for a>0,

ax =1+x\ln a+ x2(\ln a)22! + x3(\ln a)33! +…

11.3 Expansion of log(1+x)

\log(1+x) =x- x22 + x33 – x44 +…,   -1<x≤1

Here \log denotes the natural logarithm, i.e. \ln. At x=1 the alternating series converges to \ln 2.

Syllabus Note

These standard expansions are required without proof. For exam preparation, memorize their forms, conditions where applicable, and practise substitutions.

Standard Series Map Binomial / Infinite-Series Idea expand around x = 0 eˣ 1+x+x²/2!+… aˣ e^(x ln a) log(1+x) x−x²/2+x³/3−… These expansions are used for evaluation and approximation.
Figure 7: A revision map for the standard expansions named in the CDC syllabus.

12. Worked Numerical Examples

Example 1: Expand a positive integral power

Expand (2x-3)4.

(a+b)4=a4+4a3b+6a2b2+4ab3+b4.

Take a=2x, b=-3:

(2x-3)4 =16x4-96x3+216x2-216x+81.

Example 2: Find a specified coefficient

Find the coefficient of x3 in (2+x)7.

The term containing x3 occurs at r=3:

T4=7C324x3 =35×16x3=560x3.

Coefficient = 560.

Example 3: General-index expansion

Expand (1+2x)-2 up to the term in x3.

(1+y)-2 =1-2y+3y2-4y3+…

Set y=2x:

(1+2x)-2 =1-4x+12x2-32x3+…

Valid for |2x|<1, i.e. |x|<1/2.

Example 4: Approximation of a reciprocal

Approximate 1/1.02 using the binomial expansion.

(1+x)-1=1-x+x2-x3+…

Set x=0.02 and keep terms through x2:

1/1.02≈1-0.02+0.0004=0.9804.

Example 5: Approximate e0.1

Using ex=1+x+x2/2!+x3/3!+…,

e0.1≈1+0.1+0.01/2+0.001/6 =1.1051667.

Example 6: Approximate log(1.1)

Use x=0.1 in \log(1+x):

\log(1.1) ≈0.1-0.12/2+0.13/3-0.14/4

≈0.0953083.

Example 7: Sum of binomial coefficients

Evaluate 8C0+8C1+…+8C8.

Set a=x=1 in (a+x)8.

Required sum = (1+1)8=28=256.

Example 8: Alternating coefficient sum

Evaluate 7C0–7C1+7C2-…-7C7.

Set a=1, x=-1.

Required sum = (1-1)7=0.

13. Problem-Solving Strategy

  1. Identify the form: positive integral index or general index?
  2. Rewrite if necessary: for approximation, try to obtain a form (1+x)n with small |x|.
  3. Use the general term: do not fully expand if only one coefficient or one term is required.
  4. Match powers carefully: solve the exponent equation for r.
  5. Check r: in a finite positive-integral expansion, r must be an integer between 0 and n.
  6. For general index: check the convergence condition, usually |x|<1.
  7. For approximations: state the number of terms retained and keep enough terms for the required accuracy.

14. Common Mistakes and Warnings

Mistake 1: Confusing Tᵣ with Tᵣ₊₁

The term containing xr is normally Tr+1.

Mistake 2: Losing the sign

For (a-x)n, the negative sign belongs to the second term and is raised to the corresponding power.

Mistake 3: Wrong combination coefficient

Use nCr, not nPr.

Mistake 4: Forgetting convergence

General-index binomial expansions are infinite and normally require |x|<1.

Mistake 5: Too few terms in approximation

A first-order approximation may be insufficient when more accuracy is required.

Mistake 6: Treating log as base 10

In the standard expansion \log(1+x), log means natural logarithm.

15. Formula Sheet

Core Formulas
(a+x)n=Σr=0nnCran-rxr Tr+1=nCran-rxr nCr=n!/[r!(n-r)!] nCr=nCn-r nCr-1+nCr=n+1Cr (1+x)n=1+nx+n(n-1)x2/2!+… e=limn→∞(1+1/n)n ex=1+x+x2/2!+x3/3!+… ax=1+x ln a+x2(ln a)2/2!+… log(1+x)=x-x2/2+x3/3-x4/4+…

16. Important Exam Questions

Short-Answer Questions

  1. State the binomial theorem for a positive integral index.
  2. Write the general term of (a+x)n.
  3. Define a binomial coefficient and write its factorial form.
  4. How many terms are present in the expansion of (a+x)n?
  5. State one symmetry property of binomial coefficients.
  6. Write the binomial expansion of (1+x)n for a general index.
  7. State the condition under which the infinite binomial expansion is normally valid.
  8. Define Euler’s number e.
  9. Write the expansions of ex, ax and \log(1+x).

Long-Answer / Derivation Questions

  1. State and prove the binomial theorem for a positive integral index.
  2. Using the binomial theorem, derive the general term and discuss its coefficient.
  3. Prove the symmetry relation nCr=nCn-r.
  4. Use the binomial theorem to establish identities involving sums of binomial coefficients.
  5. Explain how binomial expansion is used for approximation when |x| is small.

Numerical / Application Questions

  1. Find a specified term or coefficient in a finite binomial expansion.
  2. Find the middle term(s) of a given expansion.
  3. Find the term independent of x.
  4. Expand a binomial with a fractional or negative index to a stated number of terms.
  5. Approximate roots, reciprocals or powers close to 1 by using the binomial theorem.
  6. Approximate ex, ax or \log(1+x) for a small value of x.
  7. Evaluate sums involving binomial coefficients by substituting suitable values into (a+x)n.

Diagram / Concept Questions

  1. Construct Pascal’s triangle and relate its rows to binomial coefficients.
  2. Use a factor-selection diagram to explain why the coefficient of an-rxr is nCr.
  3. Illustrate why a truncated expansion can approximate a function near x=0.

17. One-Minute Revision

Quick Revision
  • A binomial has two terms.
  • For positive integer n, (a+x)n has n+1 terms.
  • The general term is Tr+1=nCran-rxr.
  • Binomial coefficients are combinations: nCr=n!/[r!(n-r)!].
  • Coefficients are symmetric: nCr=nCn-r.
  • Pascal identity links adjacent coefficients.
  • If n is even, a finite expansion has one middle term; if n is odd, it has two.
  • For an independent term, set the total exponent of the variable equal to zero.
  • For general index, (1+x)n normally gives an infinite expansion valid for |x|<1.
  • For small |x|, higher powers are small and may be neglected for approximation.
  • e≈2.718281828 and e=lim(1+1/n)n.
  • ex=1+x+x2/2!+….
  • ax=ex\ln a.
  • \log(1+x)=x-x2/2+x3/3-….
  • Always check signs, the term number, exponent matching and convergence conditions.

18. Diagram Practice

For concept revision, practise drawing and explaining:

  • Pascal’s triangle up to at least the fifth or sixth row.
  • The exponent pattern in a finite binomial expansion.
  • The combinatorial selection idea behind the coefficient nCr.
  • The symmetry of binomial coefficients.
  • A simple graph showing why a truncated series approximates a function near x=0.
  • A conceptual sequence diagram for (1+1/n)n→e.
  • A revision map connecting ex, ax and \log(1+x).

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
Source handling: The original Nepal eNotes PDF remains embedded above. The typed section follows the verified NEB/CDC syllabus and is designed as a searchable, responsive study companion. Where the PDF viewer does not expose handwritten page text, the typed section is a syllabus-aligned reconstruction and is not claimed to be a word-for-word transcription.

Original Nepal eNotes page: Class 12 Mathematics Binomial Theorem Notes.
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