Expanding Small Binomial Powers
Simple Explanation
To expand (a+b)ⁿ for a small n, read off row n of Pascal's triangle for the coefficients, then pair each coefficient with a term aⁿ⁻ʳbʳ, where the exponent of a decreases by 1 and the exponent of b increases by 1 as you move along the row.
Why Do We Need It?
This gives a fast, reliable, mistake-resistant method for expanding a binomial power, far quicker than multiplying out (a+b)(a+b)(a+b)... by hand.
Worked Example
Expand a binomial to the 4th power
Expand (x + 2)⁴.
Why Does This Work?
Every term in the expansion of (a+b)ⁿ comes from picking either a or b from each of the n factors and multiplying — choosing b from exactly r of the n factors (and a from the rest) gives a term aⁿ⁻ʳbʳ, and there are exactly ⁿCᵣ ways to choose which r factors contribute a b, which is exactly the coefficient.
Real-Life Example
Compound interest approximations
Expanding (1+r)ⁿ for a small number of compounding periods gives each term's individual contribution to total growth.
Financial analysts sometimes expand small powers of (1+r) directly this way, to see how much each order of compounding effect contributes.
Practice
Expand (x − 1)³.
MediumCommon mistake
Forgetting to alternate signs when b is negative — each term's sign follows from bʳ itself being negative when r is odd, not from manually alternating + and − by guesswork.
Quick Review
- Use row n of Pascal's triangle for the coefficients.
- Pair each coefficient with aⁿ⁻ʳbʳ, exponents moving in opposite directions.
- If b is negative, let bʳ carry its own sign automatically — don't alternate signs by hand separately.