Binomial Expansion Calculator – (a+b)ⁿ Terms

Expanding a binomial multiplies it out systematically using binomial coefficients. Take (2x − 3)⁴: coefficients come from row 4 of Pascal’s triangle (1, 4, 6, 4, 1), the powers of 2x descend while powers of −3 ascend. Term by term: 1·(2x)⁴ = 16x⁴; 4·(2x)³(−3) = −96x³; 6·(2x)²(−3)² = 216x²; 4·(2x)(−3)³ = −216x; 1·(−3)⁴ = 81. Summed: 16x⁴ − 96x³ + 216x² − 216x + 81. The calculator prints each term before simplification so the C(n,k) weighting is visible, handles negative and fractional coefficients, and spot-checks numerically — substituting x = 1 gives 1 on both sides ((2−3)⁴ = 1 and 16−96+216−216+81 = 1 ✓). A term finder locates any requested power, and the negative-sign alternation for (a−b)ⁿ patterns is highlighted.

1. 1

   Enter both terms and the power

   e.g. 2x, −3, and 4 for (2x−3)⁴

2. 2

   Fetch row n of Pascal’s triangle

   Coefficients 1, 4, 6, 4, 1

3. 3

   Write each term

   C(n,k)·aⁿ⁻ᵏbᵏ appears before simplifying

4. 4

   Verify numerically

   Substitute a value for x on both sides

• 1

  Negative second term alternates signs across the expansion

• 2

  Term count is n+1, here 5 terms for power 4

• 3

  Numeric substitution x=1 is a 10-second correctness check

• Forgetting to raise the coefficient: (2x)³ is 8x³, not 2x³

• Losing the negative sign on odd powers of −3

• Reading 216x² from 6·(2x)²(−3)² without squaring the 3

• Dropping the final constant term (−3)⁴ = 81