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Expanding the Expression (a+b)(c+d+e)
This article explores the expansion of the expression (a+b)(c+d+e). This is a common algebraic operation that demonstrates the distributive property and is fundamental in various mathematical applications.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each term of the sum by the number separately and then adding the results. In mathematical terms:
a(b + c) = ab + ac
This property is crucial for expanding expressions like (a+b)(c+d+e).
Expanding the Expression
To expand (a+b)(c+d+e), we can apply the distributive property twice:
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Distribute (a+b) over (c+d+e): (a+b)(c+d+e) = a(c+d+e) + b(c+d+e)
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Distribute 'a' and 'b' over the terms inside the parentheses: a(c+d+e) + b(c+d+e) = ac + ad + ae + bc + bd + be
Therefore, the expanded form of (a+b)(c+d+e) is ac + ad + ae + bc + bd + be.
Applications of Expanding Expressions
Expanding expressions like (a+b)(c+d+e) has various applications in mathematics, including:
- Simplifying equations: Expanding can help simplify complex equations and make them easier to solve.
- Finding solutions: In some cases, expanding expressions can help identify solutions to equations or inequalities.
- Polynomial multiplication: This process is fundamental in polynomial multiplication and algebraic manipulations.
- Calculus: Expanding expressions is crucial in calculus when performing differentiation and integration.
Conclusion
Understanding how to expand expressions like (a+b)(c+d+e) is essential for mastering basic algebra and for further exploration of more complex mathematical concepts. The distributive property provides a powerful tool for simplifying and manipulating expressions, leading to numerous applications in various mathematical fields.