## 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.