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The Expansion of (a + b + c)³
The expansion of (a + b + c)³ is a fundamental concept in algebra, often encountered in various mathematical applications. Understanding this expansion is crucial for simplifying expressions, solving equations, and tackling problems involving polynomials.
Understanding the Process
The expansion of (a + b + c)³ can be obtained using the distributive property of multiplication and the binomial theorem. It involves multiplying the trinomial (a + b + c) by itself three times.
Here's a step-by-step breakdown:
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First Expansion: (a + b + c) * (a + b + c) = a² + ab + ac + ba + b² + bc + ca + cb + c²
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Second Expansion: (a² + ab + ac + ba + b² + bc + ca + cb + c²) * (a + b + c)
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Simplify and Combine Terms: This step requires expanding the multiplication and combining like terms. This leads to the final expanded form.
The Final Result
The expansion of (a + b + c)³ is:
a³ + 3a²b + 3a²c + 3ab² + 6abc + 3ac² + b³ + 3b²c + 3bc² + c³
Key Observations
- Symmetry: Notice the symmetry in the coefficients and terms. This pattern is a consequence of the binomial theorem.
- Coefficients: The coefficients follow a pattern related to Pascal's Triangle.
- Number of Terms: There are 10 terms in the expanded form, which corresponds to the number of ways to choose 3 elements (with repetition) from a set of 3 elements.
Applications
Understanding the expansion of (a + b + c)³ has various applications:
- Polynomial Manipulation: Expanding and simplifying expressions involving (a + b + c)³.
- Solving Equations: Using the expansion to solve cubic equations.
- Calculus: Differentiating and integrating expressions involving (a + b + c)³.
- Geometry: Applying the expansion in coordinate geometry problems.
Conclusion
The expansion of (a + b + c)³ is a fundamental concept in algebra with numerous practical applications. Mastering this expansion allows for efficient manipulation of polynomials, solving complex equations, and tackling various mathematical problems.