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Understanding the (x + y + z)² Formula: A Simple Proof
The formula (x + y + z)² = x² + y² + z² + 2xy + 2xz + 2yz might seem complex at first glance, but it's a straightforward application of the distributive property. This article will break down the proof of this formula and illustrate its application.
The Proof
The formula can be derived using the distributive property twice. We can consider (x + y + z)² as the product of (x + y + z) with itself.
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First Distribution: We begin by distributing (x + y + z) over itself:
(x + y + z)² = (x + y + z) * (x + y + z)
= x(x + y + z) + y(x + y + z) + z(x + y + z)
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Second Distribution: Now, we distribute each term within the parentheses:
= x² + xy + xz + yx + y² + yz + zx + zy + z²
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Simplification: Finally, we combine like terms and apply the commutative property of multiplication (xy = yx):
= x² + y² + z² + 2xy + 2xz + 2yz
Applications of the Formula
The (x + y + z)² formula has various applications in algebra and other fields:
- Expanding Expressions: This formula helps expand expressions involving the square of a trinomial.
- Solving Equations: It can be used to solve equations involving squared trinomials.
- Geometric Problems: The formula can be applied in geometric problems involving volumes or areas of three-dimensional objects.
Conclusion
The proof of the (x + y + z)² formula is a simple application of the distributive property and demonstrates the power of this fundamental algebraic concept. Understanding this formula allows you to manipulate and simplify expressions involving trinomials, making it a valuable tool in various mathematical contexts.