The Binomial Theorem

The binomial theorem gives a formula for expanding (x + y) raised to any whole-number power, without multiplying it out term by term. The coefficients in the expansion come straight from Pascal's triangle: row n of the triangle is exactly the set of coefficients for (x + y)ⁿ.

(x + y)ⁿ = Σ C(n, k) xⁿ⁻ₜ yₐ, summed over k = 0 to n. Each term's coefficient, C(n, k), is the same value counted by combinations — the number of ways to choose k items from n. That coefficient is also entry k of Pascal's triangle row n, so you rarely need to compute C(n, k) from scratch for small n.

To expand (x + y)³: read row 3 of Pascal's triangle — 1, 3, 3, 1 — and pair each coefficient with a term where the power of x decreases from 3 to 0 while the power of y increases from 0 to 3:

(x + y)³ = 1x³y⁰ + 3x²y¹ + 3x¹y² + 1x⁰y³ = x³ + 3x²y + 3xy² + y³

For (x − y)ⁿ or (2x + 3)ⁿ, the process is the same, but substitute the actual terms in for x and y — including their signs and coefficients — when you build each term, and simplify.

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• Algebra