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Expanding the Expression (a+b+c)(a^2+ab+bc+c^2)
This article will walk through the process of expanding the algebraic expression: (a+b+c)(a^2+ab+bc+c^2). We will use the distributive property and combine like terms to achieve the simplified form.
Step 1: Distributive Property
We begin by distributing the terms in the first parenthesis over the second parenthesis:
(a + b + c)(a² + ab + bc + c²) = a(a² + ab + bc + c²) + b(a² + ab + bc + c²) + c(a² + ab + bc + c²)
Step 2: Expanding
Now we distribute each term individually:
a(a² + ab + bc + c²) = a³ + a²b + abc + ac² b(a² + ab + bc + c²) = a²b + ab² + b²c + bc² c(a² + ab + bc + c²) = a²c + abc + b²c + c³
Step 3: Combining Like Terms
Finally, we combine the terms with the same variables and exponents:
a³ + a²b + abc + ac² + a²b + ab² + b²c + bc² + a²c + abc + b²c + c³ = a³ + 2a²b + 2abc + ab² + 2b²c + ac² + bc² + c³
Conclusion
Therefore, the expanded form of the expression (a+b+c)(a² + ab + bc + c²) is: a³ + 2a²b + 2abc + ab² + 2b²c + ac² + bc² + c³. This process demonstrates how the distributive property can be used to expand complex algebraic expressions and simplify them by combining like terms.