CBSE Math RevisionBinomial Theorem: CBSE Class 11 Maths knowledge organiser
Everything to know about binomial theorem on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.
Key definitions
- Binomial theorem
- Expands (a + b)ⁿ for a positive integer n using the coefficients ⁿCᵣ.
- General term
- Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ.
- Pascal's triangle
- Each entry is the sum of the two above it; row n gives the coefficients of (a + b)ⁿ.
Key formulas
| Expansion | \((a+b)^n=\sum_{r=0}^n{}^nC_ra^{n-r}b^r\) |
| \(n+1\) terms; sum of coefficients | \(\sum{}^nC_r=2^n,\) \(\sum(-1)^r{}^nC_r=0\) |
Worked example
Expand \((1 + 2x)^3\).
\(1 + 3(2x) + 3(2x)^2 + (2x)^3 = 1 + 6x + 12x^2 + 8x^3\).
Common mistakes
- Forgetting to raise the coefficient of \(x\) to the power: \((2x)^3 = 8x^3\), not \(2x^3\).
- Signs in \((a - b)^n\): every odd power of \(b\) carries a minus sign.
- Using row \(n - 1\) of Pascal's triangle for \((a + b)^n\) (the row starting \(1, n\) is the right one).
- Writing \(n\) terms instead of \(n + 1\).
You should be able to…
- Use Pascal's triangle, count terms and find coefficients in small expansions.
- Expand binomials with coefficients and fractions and evaluate powers of numbers near round numbers.
- Prove divisibility results and combine (a + b)ⁿ ± (a − b)ⁿ in multi-part questions.
- Compare sizes of powers and reason about sums of coefficients.
The printable sheet
Revise it next
- Binomial Theorem: Class 11 notes
- Practise Binomial Theorem by step (Route to 95)
- Skill Builders
- CBSE Class 11 Maths formula sheet (PDF)
Other CBSE Class 11 Maths topics: Sets · Relations and Functions · Trigonometric Functions · Complex Numbers and Quadratic Equations · Linear Inequalities · Permutations and Combinations · Sequences and Series · Straight Lines · Conic Sections · Introduction to Three Dimensional Geometry · Limits and Derivatives · Statistics · Probability · All CBSE Class 11 Maths organisers