%5E3-expanded-form.jpeg "(x+y)^3 Expanded Form")
The Expansion of (x + y)³
The expansion of (x + y)³ is a fundamental concept in algebra and finds frequent application in various mathematical fields. Understanding this expansion is crucial for simplifying expressions, solving equations, and gaining a deeper grasp of polynomial operations.
Expanding the Expression
To expand (x + y)³, we can utilize the distributive property of multiplication, applying it repeatedly. Here's the breakdown:
(x + y)³ = (x + y)(x + y)(x + y)
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First Expansion: (x + y)(x + y) = x(x + y) + y(x + y) = x² + xy + xy + y² = x² + 2xy + y²
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Second Expansion: (x² + 2xy + y²)(x + y) = x²(x + y) + 2xy(x + y) + y²(x + y) = x³ + x²y + 2x²y + 2xy² + xy² + y³ = x³ + 3x²y + 3xy² + y³
Therefore, the expanded form of (x + y)³ is x³ + 3x²y + 3xy² + y³.
Understanding the Pattern
Notice the pattern emerging in the expanded form:
- Coefficients: The coefficients follow a specific pattern: 1, 3, 3, 1. These numbers are the binomial coefficients, which can be obtained from Pascal's Triangle.
- Exponents: The exponents of x decrease from 3 to 0, while the exponents of y increase from 0 to 3.
This pattern allows us to expand any power of (x + y) without repeatedly applying the distributive property.
Applications
The expansion of (x + y)³ has several applications, including:
- Factoring Polynomials: Knowing the expanded form helps in recognizing and factoring expressions that contain the pattern of (x + y)³.
- Solving Equations: Expanding (x + y)³ can simplify equations, making them easier to solve.
- Calculus: The expansion is crucial in differentiating and integrating expressions.
Conclusion
The expansion of (x + y)³ is a key concept in algebra, with applications across various mathematical fields. Understanding the pattern and its applications empowers you to simplify expressions, solve equations, and deepen your understanding of polynomial manipulation.