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Simplifying the Expression (x⁴ - y⁴) / (x - y)
This article will explore the simplification of the algebraic expression (x⁴ - y⁴) / (x - y). We will use a combination of algebraic identities and factoring techniques to arrive at a concise and simplified form.
Understanding the Expression
The expression (x⁴ - y⁴) / (x - y) represents a fraction where the numerator is the difference of two fourth powers and the denominator is the difference of two variables. Our goal is to simplify this expression to a form that is easier to work with and understand.
Applying the Difference of Squares Identity
We can begin simplifying the expression by recognizing that the numerator (x⁴ - y⁴) can be factored using the difference of squares identity:
a² - b² = (a + b)(a - b)
Applying this to our numerator:
x⁴ - y⁴ = (x²)² - (y²)² = (x² + y²)(x² - y²)
Now our expression becomes:
(x² + y²)(x² - y²) / (x - y)
Factoring Further
Notice that the term (x² - y²) in the numerator is again a difference of squares. Applying the same identity again:
(x² - y²) = (x + y)(x - y)
Substituting this back into our expression:
(x² + y²)(x + y)(x - y) / (x - y)
Simplifying the Expression
Finally, we can cancel out the common factor of (x - y) in the numerator and denominator:
(x² + y²)(x + y)
Result
The simplified form of the expression (x⁴ - y⁴) / (x - y) is (x² + y²)(x + y).
This simplification demonstrates the power of recognizing and applying algebraic identities in simplifying complex expressions.