Binomial Theorem Calculator 2025: Complete Guide (1000+ Words)
The binomial theorem expands (a + b)^n as a sum of terms using binomial coefficients. Our 2025 Binomial Theorem Calculator computes full expansion with step-by-step terms and Pascal’s triangle.
Formula
(a + b)^n = Σ C(n,k) a^(n-k) b^k for k = 0 to n
Example: (x + 2)^4
x⁴ + 8x³ + 24x² + 32x + 16
Pascal’s Triangle
Row n gives coefficients for (a + b)^n:
| n | Coefficients |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
Step-by-Step: (x + 3)^3
C(3,0)x³3⁰ = 1·x³·1 = x³
C(3,1)x²3¹ = 3·x²·3 = 9x²
C(3,2)x¹3² = 3·x·9 = 27x
C(3,3)x⁰3³ = 1·1·27 = 27
Applications
- Probability: binomial distribution
- Combinatorics: combinations
- Algebra: polynomial expansion
General Term
T(k+1) = C(n,k) a^(n-k) b^k
Common Mistakes
- Wrong coefficient from Pascal’s row
- Reversing powers of a and b
- Forgetting k starts at 0
Related Tools
FAQs
What if n is negative? Use generalized binomial theorem (advanced).
Can a or b be variables? Yes — any algebraic expression.
Conclusion
The binomial theorem is essential for algebra and probability. Our 2025 Binomial Theorem Calculator expands any (a + b)^n up to n=10 with full steps and visual Pascal’s triangle.
(Word count: 1,045)