%5E2.jpeg "(a+b+c+d)^2")
Expanding (a + b + c + d)^2
The expression (a + b + c + d)^2 represents the square of the sum of four variables. Expanding this expression involves multiplying the sum by itself.
Using the Distributive Property
The most straightforward way to expand this expression is to use the distributive property repeatedly. Here's how:
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First, expand the square: (a + b + c + d)^2 = (a + b + c + d) * (a + b + c + d)
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Apply the distributive property to the first term: a * (a + b + c + d) + b * (a + b + c + d) + c * (a + b + c + d) + d * (a + b + c + d)
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Distribute each term further: a^2 + ab + ac + ad + ba + b^2 + bc + bd + ca + cb + c^2 + cd + da + db + dc + d^2
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Combine like terms: a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
General Formula
The expansion of (a + b + c + d)^2 can be generalized as follows:
The square of the sum of four variables is equal to the sum of the squares of each variable plus twice the product of each pair of variables.
This formula can be applied to any number of variables. For example, the expansion of (a + b + c)^2 would be:
a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
Applications
Expanding (a + b + c + d)^2 is a fundamental concept in algebra and has applications in various areas, including:
- Polynomial simplification: Expanding expressions can make them easier to manipulate and simplify.
- Solving equations: Expanding squares can be helpful in solving equations involving multiple variables.
- Calculus: Understanding the expansion of squares is crucial for differentiation and integration.
By understanding the expansion of (a + b + c + d)^2, you gain a valuable tool for working with algebraic expressions.