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Binomial theorem

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The Binomial Theorem

The binomial theorem expands (x+y)^n as a sum of terms whose coefficients come from Pascal's triangle. Learn the general formula (x+y)^n = the sum of C(n,k) x^(n-k) y^k, and see it applied to a full worked expansion of (x+y) cubed.

What the binomial theorem says

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.

Binomial theorem: expanding (x + y)^3 (x+y)^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3. The coefficients 1, 3, 3, 1 are exactly row 3 of Pascal's triangle. (x + y)³ = 1x³ + 3x²y + 3xy² + 1y³ coefficients 1, 3, 3, 1 = Pascal's triangle row 3 General form: (x+y)ⁿ = Σ C(n,k) xⁿ⁻ₜ yₐ
Expanding (x + y)³ using row 3 of Pascal's triangle: 1, 3, 3, 1.

The general formula

(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.

Worked example

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

Handling a minus sign or a coefficient

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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