(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Jasonbradley

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Jun 16, 2024

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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

1. 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)

2. 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

3. 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.