Binomial theorem
Expanding (a + b)ⁿ with Pascal’s triangle and ⁿCᵣ.
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To expand (a + b)ⁿ, take the numbers from Pascal’s triangle.
Look: Row 4 (yellow): 1, 4, 6, 4, 1.
✏️ Your turn!
Type your answer and press Check. Kutty will tell you if it is right.
1Row 5 of Pascal’s triangle?
2How many terms in (a + b)⁶?
3Expand (x + 2)²
! Careful!
✗(a + b)² = a² + b².
✓Row 2 is 1, 2, 1: a² + 2ab + b².
✗Forgetting to power the number: (2x)³ = 2x³.
✓(2x)³ = 8x³.
See the whole lesson on one page
To expand (a + b)ⁿ, take the numbers from Pascal’s triangle.
Row 4 of the triangle is 1, 4, 6, 4, 1. So (a + b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴. The power of a goes down while b goes up.
Think of it like thisPascal’s triangle is built like a stack of bricks: each number is the two above it added together.
Solved together
Expand (x + 1)³.
Row 3 of Pascal’s triangle: 1, 3, 3, 1.
Powers of x go down 3, 2, 1, 0. Powers of 1 are just 1.
x³ + 3x² + 3x + 1
Answer(x + 1)³ = x³ + 3x² + 3x + 1.
For older learners and grown-ups: a little more
General term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. Coefficient of x⁵ in (x + 2)⁸: r = 3, so ⁸C₃ × 2³ = 56 × 8 = 448.
Now practise it
Kanakku makes new questions on binomial theorem every time. It shows the working and gives a hint when you are stuck.