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Expanding a binomial fully: Class 11 practice

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + … + ⁿCₙbⁿ. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • ⁿCᵣ
  • Laws of exponents

The idea behind expanding a binomial fully

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + … + ⁿCₙbⁿ. Coefficients come from Pascal's triangle; powers of a fall while powers of b rise, always adding to n.

Step-by-step method

  1. Write the coefficients ⁿC₀ … ⁿCₙ.
  2. Attach decreasing powers of a and increasing powers of b.
  3. Simplify each term (signs for a − b).

Worked example 1 (Basic)

Expand (x + 1)^3.

  1. Coefficients 1, 3, 3, 1
  2. x^3 + 3x^2 + 3x + 1
  3. Answer: x^3 + 3x^2 + 3x + 1

Worked example 2 (Standard)

Expand (x + 2)^4.

  1. Coefficients 1, 4, 6, 4, 1
  2. x^4 + 8x^3 + 24x^2 + 32x + 16
  3. Answer: x^4 + 8x^3 + 24x^2 + 32x + 16

Worked example 3 (Challenge)

Expand (x − 3)^5.

  1. Coefficients 1, 5, 10, 10, 5, 1
  2. x^5 − 15x^4 + 90x^3 − 270x^2 + 405x − 243
  3. Answer: x^5 − 15x^4 + 90x^3 − 270x^2 + 405x − 243

Common mistakes and misconceptions

  • Forgetting to raise the coefficient of x
  • Sign pattern errors for (a − b)ⁿ

Practice questions

  1. (Basic) Expand (x + 1)^4.
  2. (Basic) Expand (x + 3)^3.
  3. (Standard) Expand (x + 1)^5.
  4. (Standard) Expand (x + 3)^5.
  5. (Challenge) Expand (x − 1)^5.
  6. (Challenge) Expand (x − 1)^4.

Answer key with full working

  • Q1: Coefficients 1, 4, 6, 4, 1 → x^4 + 4x^3 + 6x^2 + 4x + 1 → Answer: x^4 + 4x^3 + 6x^2 + 4x + 1
  • Q2: Coefficients 1, 3, 3, 1 → x^3 + 9x^2 + 27x + 27 → Answer: x^3 + 9x^2 + 27x + 27
  • Q3: Coefficients 1, 5, 10, 10, 5, 1 → x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1 → Answer: x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1
  • Q4: Coefficients 1, 5, 10, 10, 5, 1 → x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243 → Answer: x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243
  • Q5: Coefficients 1, 5, 10, 10, 5, 1 → x^5 − 5x^4 + 10x^3 − 10x^2 + 5x − 1 → Answer: x^5 − 5x^4 + 10x^3 − 10x^2 + 5x − 1
  • Q6: Coefficients 1, 4, 6, 4, 1 → x^4 − 4x^3 + 6x^2 − 4x + 1 → Answer: x^4 − 4x^3 + 6x^2 − 4x + 1

Syllabus note

This skill is listed under "Binomial Theorem" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is expanding a binomial fully in?

"Binomial Theorem", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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