(a%5E2%2Bb%5E2%2Bc%5E2-ab-bc-ca).jpeg "(a+b+c)(a^2+b^2+c^2-ab-bc-ca)")
Expanding the Expression: (a+b+c)(a^2+b^2+c^2-ab-bc-ca)
This expression represents a fundamental concept in algebra, showcasing the expansion of a product involving three variables. Let's explore how to expand this expression and uncover its interesting properties.
Step-by-Step Expansion
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Distribute: We start by distributing the first term, (a+b+c), to each term within the second parentheses:
(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a(a^2+b^2+c^2-ab-bc-ca) + b(a^2+b^2+c^2-ab-bc-ca) + c(a^2+b^2+c^2-ab-bc-ca)
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Simplify: Now, we multiply each term inside the parentheses by the corresponding coefficient:
= a^3 + ab^2 + ac^2 - a^2b - abc - a^2c + ba^2 + b^3 + bc^2 - ab^2 - b^2c - abc + ca^2 + cb^2 + c^3 - abc - bc^2 - c^2a
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Combine Like Terms: Finally, we combine terms with the same variables and exponents:
= a^3 + b^3 + c^3 - 3abc
Key Observation
The expanded form of (a+b+c)(a^2+b^2+c^2-ab-bc-ca) simplifies to a^3 + b^3 + c^3 - 3abc. This is a noteworthy result with applications in various mathematical contexts.
Applications and Significance
This expression holds relevance in areas like:
- Factorization: It provides a shortcut for factoring a^3 + b^3 + c^3 - 3abc.
- Algebraic Manipulation: Understanding the expansion helps manipulate complex expressions involving three variables.
- Geometric Interpretation: It has connections to the geometry of cubes and the volume of a parallelepiped.
Conclusion
The expansion of (a+b+c)(a^2+b^2+c^2-ab-bc-ca) reveals a valuable algebraic identity. Its simplicity and the elegance of the result highlight the power of algebraic manipulation and provide a foundation for understanding more complex concepts in mathematics.