PanMaths.comFree Printable Math Formula Reference Sheet
Binomial Theorem
Algebra • Grades 10, 11, 12
Algebra
Binomial Theorem
Expand binomials raised to any power.
The Formula
(a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ
Used for: Algebraically expanding binomial expressions (a + b)ⁿ raised to any positive integer power n
a, bFirst and second terms of the binomial
nPositive integer exponent power
C(n, k)Binomial coefficient: n! / (k!(n-k)!)
⚡Live Interactive Quick Solver
Instant Calculation(x + 1)^3 =x^3 +3x^2 +3x +1
Pascal's Row (n=3): [1, 3, 3, 1]• 4 terms in the expansion
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
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🎯 Key Concepts
Understanding binomial expansion:
🔢
C(n,k) = n! / (k!(n-k)!)Binomial coefficient formula
📐
Pascal's TriangleQuick way to find coefficients
🔄
n+1 terms totalExpansion has one more term than the power
📝 Worked Examples
Expand (x+1)³
Expand:
(x+1)³ = ?1
Pascal's row for n=3
1, 3, 3, 12
Apply powers
1·x³ + 3·x²·1 + 3·x·1² + 1·1³3
Simplify
x³ + 3x² + 3x + 1Answer: x³ + 3x² + 3x + 1
Expand (2x-1)⁴
Expand:
(2x-1)⁴ = ?1
Pascal's row for n=4
1, 4, 6, 4, 12
First terms
1(2x)⁴ + 4(2x)³(-1) + 6(2x)²(-1)²...3
Simplify
16x⁴ - 32x³ + 24x² - 8x + 1Answer: 16x⁴ - 32x³ + 24x² - 8x + 1
💡 Pro Tips
Pascal's Triangle
Row n: 1, n, n(n-1)/2, ... — each entry is sum of two above it
Signs alternate
For (a-b)ⁿ, signs alternate: +, -, +, -, ...
First & last terms
First term: aⁿ, Last term: bⁿ
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