Expanding the Expression: (a+b+c)(a^2+b^2+c^2abbcca)
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.
StepbyStep Expansion

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^2abbcca) = a(a^2+b^2+c^2abbcca) + b(a^2+b^2+c^2abbcca) + c(a^2+b^2+c^2abbcca)

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

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^2abbcca) 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^2abbcca) 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.