%5E3-proof.jpeg "(a+b+c)^3 Proof")
Expanding (a + b + c)^3: A Step-by-Step Proof
The expansion of (a + b + c)^3 can be a daunting task, but it's essential to understand the process and its implications. Here's a breakdown of how to prove the expansion:
Understanding the Concept
(a + b + c)^3 essentially means multiplying (a + b + c) by itself three times:
(a + b + c) * (a + b + c) * (a + b + c)
The Expansion Process
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First Multiplication: Start by multiplying the first two factors: (a + b + c) * (a + b + c) = a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2
Remember that multiplication is commutative, so ab = ba, ac = ca, and bc = cb.
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Simplifying: Combine like terms: a^2 + 2ab + 2ac + 2bc + b^2 + c^2
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Second Multiplication: Multiply the result from step 2 by the third factor (a + b + c): (a^2 + 2ab + 2ac + 2bc + b^2 + c^2) * (a + b + c)
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Expanding: Apply the distributive property: a^3 + 2a^2b + 2a^2c + 2abc + ab^2 + ac^2 + 2ab^2 + 4abc + 4abc + 2b^2c + 2ac^2 + 4abc + 2b^2c + 2bc^2 + b^3 + c^3
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Combining Like Terms: Finally, combine the coefficients of identical terms: a^3 + 3a^2b + 3a^2c + 6abc + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + b^3 + c^3
The Final Result
Therefore, the expansion of (a + b + c)^3 is:
a^3 + 3a^2b + 3a^2c + 6abc + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + b^3 + c^3
Key Points
- The Binomial Theorem: The expansion of (a + b + c)^3 can be derived using the Binomial Theorem, which provides a general formula for expanding any binomial raised to a power.
- Symmetry: Notice the symmetry in the expansion. The terms are symmetrical with respect to a, b, and c.
- Applications: This expansion is crucial in various fields, including algebra, calculus, and statistics.
By understanding the steps involved, you can confidently expand (a + b + c)^3 and apply its result in various mathematical contexts.