Binomial Expansion Calculator — Expand (a + b)^n Instantly
Expand binomial expressions using the binomial theorem. For completing the square, try our completing the square calculator.
Table of Contents
Expand a Binomial
Enter the binomial expression and the power n.
Binomial Expansion – Complete Explanation
Binomial expansion expands (a + b)^n into a sum of terms using combinations.
Binomial Theorem
(a + b)ⁿ = Σ [ⁿCᵣ × aⁿ⁻ʳ × bʳ] for r = 0 to n
Where ⁿCᵣ = n! / (r! × (n-r)!)
1. What Is a Binomial Expansion?
A binomial has exactly two terms. Expansion raises it to a power.
(a + b)² = a² + 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
2. The Binomial Theorem Formula
Each term has:
- ⁿCᵣ: Binomial coefficient.
- aⁿ⁻ʳ: First term, decreasing power.
- bʳ: Second term, increasing power.
3. Pascal's Triangle
Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1
4. Combinations (nCr) Explained
Combination Formula
ⁿCᵣ = n! / (r! × (n-r)!)
⁵C₂ = 5!/(2!×3!) = 120/12 = 10
5. Step-by-Step Examples
5.1 Example 1: (x + y)⁵
Coefficients: 1, 5, 10, 10, 5, 1 (x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵
5.2 Example 2: (2x - 3)⁴
Coefficients: 1, 4, 6, 4, 1 (2x - 3)⁴ = 16x⁴ - 96x³ + 216x² - 216x + 81
6. More Examples
(1 + x)³ = 1 + 3x + 3x² + x³
(a - b)⁶ = a⁶ - 6a⁵b + 15a⁴b² - 20a³b³ + 15a²b⁴ - 6ab⁵ + b⁶
(3x + 2y)³ = 27x³ + 54x²y + 36xy² + 8y³
7. Patterns in Binomial Expansions
- n + 1 terms
- Exponents sum to n in each term
- Symmetric coefficients
- Alternating signs when b is negative
8. Negative and Fractional Exponents
(1 + x)⁻¹ = 1 - x + x² - x³ + x⁴ - ... (for |x| < 1)
9. Common Mistakes to Avoid
- Forgetting binomial coefficients — not just the variables.
- Misplacing exponents — exponents always sum to n.
- Ignoring signs — negative terms alternate signs.
- Not raising coefficients — (2x)⁴ = 16x⁴, not 2x⁴.
10. Real-World Applications
- Probability: Binomial distribution.
- Compound Interest: (1 + r)ⁿ expansion.
- Calculus: Series approximations.
- Combinatorics: Counting subsets.
11. Practice Problems with Solutions
(x + 2)³ = x³ + 6x² + 12x + 8
(a - 1)⁴ = a⁴ - 4a³ + 6a² - 4a + 1
Coefficient of x⁵ in (1+x)⁸ = 56
(2x - y)³ = 8x³ - 12x²y + 6xy² - y³
12. Tips for Mastering Binomial Expansion
- Memorize Pascal's triangle through row 7.
- Use symmetry of coefficients.
- Check: number of terms = n + 1.
- Verify exponents sum to n in each term.
13. Final Thoughts
The binomial theorem is one of the most elegant and useful results in algebra. Use this calculator to verify your work, but practice manually to build confidence.
14. Frequently Asked Questions
Use the combination formula: ⁿCᵣ = n! / (r! × (n-r)!). For example, ⁷C₃ = 7!/(3!×4!) = 5040/(6×24) = 35. You can also use the shortcut: ⁷C₃ = (7×6×5)/(3×2×1) = 210/6 = 35.
Each term in the expansion is a product of factors chosen from the n factors of (a + b). In each term, you pick 'a' (n-r) times and 'b' r times, for a total of (n-r) + r = n factors. The exponents reflect how many times each variable is chosen, so they must sum to n.
The binomial distribution uses binomial coefficients to calculate P(k successes in n trials) = ⁿCₖ × pᵏ × (1-p)ⁿ⁻ᵏ. The coefficients ⁿCₖ come directly from the binomial expansion, making the theorem fundamental to probability theory.
Yes, but the expansion becomes an infinite series rather than a finite sum. For example, (1 + x)⁻¹ = 1 - x + x² - x³ + x⁴ - ... for |x| < 1. This is called the generalized binomial theorem and requires convergence conditions.
Each entry in Pascal's triangle equals a binomial coefficient. Row n, position r (starting from 0) equals ⁿCᵣ. Pascal's triangle provides a visual, recursive way to compute combinations without factorial calculations. Each number is the sum of the two above it.
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Conclusion
The binomial theorem is one of the most elegant and useful results in algebra. Use this calculator to verify your work, but practice manually to build confidence.
Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.