(x + Y)^3

3 min read Jun 16, 2024

The Cube of a Binomial: (x + y)³

In algebra, (x + y)³ represents the cube of a binomial, which is a polynomial with two terms, x and y. This expression can be expanded using the binomial theorem or by applying the distributive property. Let's explore both methods:

Expanding using the Binomial Theorem

The binomial theorem provides a general formula for expanding any binomial raised to a power:

(x + y)ⁿ = ∑(n choose k) * xⁿ⁻ᵏ * yᵏ

Where:

n is the power to which the binomial is raised.
k ranges from 0 to n.
(n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).

Applying this to (x + y)³, we get:

(x + y)³ = (3 choose 0) * x³ * y⁰ + (3 choose 1) * x² * y¹ + (3 choose 2) * x¹ * y² + (3 choose 3) * x⁰ * y³

Simplifying the binomial coefficients:

(x + y)³ = x³ + 3x²y + 3xy² + y³

Expanding using the Distributive Property

We can also expand (x + y)³ by repeatedly applying the distributive property:

(x + y)³ = (x + y) * (x + y) * (x + y)

First, expand the first two factors:

(x + y) * (x + y) = x² + 2xy + y²

Now, multiply this result by (x + y):

(x² + 2xy + y²) * (x + y) = x³ + 2x²y + xy² + x²y + 2xy² + y³

Combining like terms:

(x + y)³ = x³ + 3x²y + 3xy² + y³

Key Observations

The expansion always results in four terms.
The coefficients of the terms follow the pattern of Pascal's Triangle.
The exponents of x decrease from 3 to 0, while the exponents of y increase from 0 to 3.

Applications

The expansion of (x + y)³ has various applications in different areas of mathematics, including:

Calculus: Finding derivatives and integrals of functions involving binomials.
Algebra: Solving equations and simplifying expressions.
Statistics: Deriving formulas for statistical measures.

Understanding the expansion of (x + y)³ provides a solid foundation for working with more complex expressions and solving problems in various fields.