Expanding (a + b + c + d + e)²: A Comprehensive Guide
The expansion of (a + b + c + d + e)² is a common algebraic operation with numerous applications in mathematics and other fields. While the formula itself might seem intimidating, understanding the underlying process makes it much easier to work with. Let's break it down step by step:
Understanding the Concept
The expression (a + b + c + d + e)² essentially represents the product of the entire sum with itself:
(a + b + c + d + e)² = (a + b + c + d + e) * (a + b + c + d + e)
To expand this, we need to multiply each term in the first bracket with each term in the second bracket. This process involves several multiplications, leading to a multitude of terms.
Applying the Distributive Property
We can systematically expand the expression by applying the distributive property repeatedly. This means we multiply each term in the first bracket with all the terms in the second bracket:

a * (a + b + c + d + e): This gives us a² + ab + ac + ad + ae.

b * (a + b + c + d + e): This results in ab + b² + bc + bd + be.

c * (a + b + c + d + e): Expanding this gives ac + bc + c² + cd + ce.

d * (a + b + c + d + e): This gives ad + bd + cd + d² + de.

e * (a + b + c + d + e): Finally, we get ae + be + ce + de + e².
Combining Like Terms
Now, we have a long list of terms. Notice that many of them are repeated. To simplify the expression, we combine like terms:
(a + b + c + d + e)² = a² + b² + c² + d² + e² + 2ab + 2ac + 2ad + 2ae + 2bc + 2bd + 2be + 2cd + 2ce + 2de
The General Formula
We can generalize the formula for any number of terms:
(x₁ + x₂ + ... + xₙ)² = x₁² + x₂² + ... + xₙ² + 2(x₁x₂ + x₁x₃ + ... + x₁xₙ + x₂x₃ + ... + x₂xₙ + ... + xₙ₋₁xₙ)
Example: Applying the Formula
Let's take a concrete example to see how this works in practice:
Suppose we want to expand (2 + 3 + 4 + 5)².
Applying the formula:
(2 + 3 + 4 + 5)² = 2² + 3² + 4² + 5² + 2(23 + 24 + 25 + 34 + 35 + 45)
Simplifying:
(2 + 3 + 4 + 5)² = 4 + 9 + 16 + 25 + 2(6 + 8 + 10 + 12 + 15 + 20)
(2 + 3 + 4 + 5)² = 49 + 2(71)
(2 + 3 + 4 + 5)² = 49 + 142
(2 + 3 + 4 + 5)² = 191
Conclusion
Expanding (a + b + c + d + e)² can be easily done by applying the distributive property and combining like terms. The general formula allows you to handle expressions with any number of terms, making it a powerful tool for algebraic manipulation.