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Simplifying the Expression: (x² + y² - z² + 2xy) / (x² - y² - z² - 2yz)
This article explores the simplification of the algebraic expression: (x² + y² - z² + 2xy) / (x² - y² - z² - 2yz)
Recognizing Patterns and Factoring
The key to simplifying this expression lies in recognizing the patterns within the numerator and denominator. We can utilize the following algebraic identities:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
By applying these identities, we can rewrite the expression as:
Numerator: (x² + y² - z² + 2xy) = (x + y)² - z² Denominator: (x² - y² - z² - 2yz) = (x - z)² - y²
Applying the Difference of Squares
We now have both numerator and denominator in the form of a difference of squares: a² - b² = (a + b)(a - b). Applying this:
Numerator: (x + y)² - z² = (x + y + z)(x + y - z) Denominator: (x - z)² - y² = (x - z + y)(x - z - y)
Final Simplification
Combining the simplified numerator and denominator, the expression becomes:
(x + y + z)(x + y - z) / (x - z + y)(x - z - y)
This is the simplified form of the original expression. While it may not look simpler at first glance, it is now factored and easier to analyze and manipulate.
Importance of Factoring
Factoring is a crucial technique in algebra for simplifying expressions and solving equations. It allows us to:
- Identify common factors: This can lead to cancellations, simplifying the expression.
- Solve for unknowns: Factoring allows us to rewrite equations in a form where we can easily isolate the unknown variables.
- Analyze relationships: Understanding the factored form can reveal underlying relationships and patterns within the expression.
In this case, factoring the expression allowed us to simplify it and make it easier to work with in further calculations or analyses.