Here is the expansion of (x + y)ⁿ for n = 0, 1,…, 5:

(x + y)⁰ = 1
(x + y)¹ = x + y
(x + y)² = x² +2xy + y²
(x + y)³ = x³ +3x²y + 3xy² + y³
(x + y)⁴ = x⁴ +4x³y + 6x²y² +4xy³ + y⁴
(x + y)⁵ = x⁵ +5x⁴y + 10x³y² +10x²y³ +5xy⁴ + y⁵

Look familiar? The coefficients of each expansion are the entries in Row n of Pascal's Triangle. Thus, the coefficient of each term r of the expansion of (x + y)ⁿ is given by C(n, r - 1). The exponents of x descend, starting with n, and the exponents of y ascend, starting with 0, so the r^th term of the expansion of (x + y)² contains x^n-(r-1)y^r-1.

This information can be summarized by the Binomial Theorem:

Example: Write out the expansion of (x + y)⁷.

(x + y)⁷ = x⁷ +7x⁶y + 21x⁵y² +35x⁴y³ +35x³y⁴ +21x²y⁵ +7xy⁶ + y⁷.

When the terms of the binomial have coefficient(s), be sure to apply the exponents to these coefficients.

Example: Write out the expansion of (2x + 3y)⁴.

Example: Write out the expansion of (5x - y)³.