%5E3-expansion.jpeg "(a+b)^3 Expansion")
Expanding (a + b)³: A Step-by-Step Guide
The expansion of (a + b)³ is a fundamental concept in algebra, often encountered in various mathematical problems. Understanding this expansion is essential for simplifying expressions, solving equations, and tackling more complex algebraic manipulations.
Understanding the Concept
(a + b)³ is simply (a + b) multiplied by itself three times:
(a + b)³ = (a + b)(a + b)(a + b)
To expand this expression, we can use the distributive property (or FOIL method) multiple times.
Step-by-Step Expansion
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Expand the first two factors: (a + b)(a + b) = a(a + b) + b(a + b) = a² + ab + ba + b² = a² + 2ab + b²
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Multiply the result by (a + b): (a² + 2ab + b²)(a + b) = a(a² + 2ab + b²) + b(a² + 2ab + b²) = a³ + 2a²b + ab² + ba² + 2ab² + b³ = a³ + 3a²b + 3ab² + b³
The Final Result
Therefore, the expansion of (a + b)³ is:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Key Observations
- Coefficients: The coefficients of the terms follow a specific pattern: 1, 3, 3, 1. This pattern is related to the binomial theorem and Pascal's Triangle.
- Exponents: The exponents of 'a' decrease from 3 to 0, while the exponents of 'b' increase from 0 to 3.
Applications
The expansion of (a + b)³ has numerous applications in various fields, including:
- Algebraic manipulation: Simplifying complex expressions and solving equations.
- Calculus: Finding derivatives and integrals.
- Physics: Analyzing motion and forces.
- Statistics: Calculating probabilities and distributions.
Conclusion
Understanding the expansion of (a + b)³ is crucial for mastering basic algebra and building a strong foundation for more advanced mathematical concepts. By utilizing the distributive property and recognizing the pattern in the coefficients and exponents, we can efficiently expand this expression and apply it in various contexts.