%5E3-expand.jpeg "(x+y)^3 Expand")
Understanding the Expansion of (x + y)³
The expression (x + y)³ represents the cube of the binomial (x + y). Expanding this expression involves multiplying the binomial by itself three times.
Expanding the Expression
There are two common methods to expand (x + y)³:
1. Direct Multiplication:
- Step 1: Multiply (x + y) by (x + y). (x + y) * (x + y) = x² + 2xy + y²
- Step 2: Multiply the result from step 1 by (x + y) again. (x² + 2xy + y²) * (x + y) = x³ + 3x²y + 3xy² + y³
2. Using the Binomial Theorem:
The binomial theorem provides a general formula for expanding any power of a binomial. For (x + y)³, the theorem gives:
(x + y)³ = ¹C₀ x³ y⁰ + ¹C₁ x² y¹ + ¹C₂ x¹ y² + ¹C₃ x⁰ y³
Where ¹C₀, ¹C₁, ¹C₂, and ¹C₃ are binomial coefficients, which can be calculated using the formula:
nCr = n! / (r! * (n-r)!)
Applying this to our expression:
- ¹C₀ = 1! / (0! * 1!) = 1
- ¹C₁ = 1! / (1! * 0!) = 1
- ¹C₂ = 1! / (2! * -1!) = 1
- ¹C₃ = 1! / (3! * -2!) = 1
Substituting these values back into the binomial theorem:
(x + y)³ = x³ + 3x²y + 3xy² + y³
Key Observations
- The coefficients: Notice that the coefficients of the expanded expression follow a pattern: 1, 3, 3, 1. This pattern is known as Pascal's Triangle.
- The exponents: The exponents of x decrease from 3 to 0, while the exponents of y increase from 0 to 3.
Applications
Understanding the expansion of (x + y)³ is crucial in various mathematical areas:
- Algebra: Simplifying expressions, solving equations, and factoring polynomials.
- Calculus: Finding derivatives and integrals of functions involving binomial expressions.
- Statistics: Analyzing data and calculating probabilities.
- Physics: Modeling physical phenomena that involve power relationships.
Conclusion
The expansion of (x + y)³ is a fundamental concept in mathematics with numerous applications. By understanding the direct multiplication and binomial theorem methods, we can effectively expand this expression and utilize it for various purposes.