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Factoring the Difference of Squares: (y-8)(y+8)
The expression (y-8)(y+8) is a special case of factoring known as the difference of squares. This pattern is frequently encountered in algebra and understanding it can greatly simplify calculations.
The Difference of Squares Pattern
The difference of squares pattern states:
(a - b)(a + b) = a² - b²
This pattern holds true because when we expand the left side of the equation, the middle terms cancel out:
- (a - b)(a + b) = a(a + b) - b(a + b)
- = a² + ab - ba - b²
- = a² - b²
Applying the Pattern to (y-8)(y+8)
In our expression, (y-8)(y+8), we can identify:
- a = y
- b = 8
Applying the difference of squares pattern, we get:
- (y - 8)(y + 8) = y² - 8²
Simplifying further:
- y² - 8² = y² - 64
Therefore, the factored form of (y-8)(y+8) is y² - 64.
Importance of Recognizing the Pattern
Recognizing the difference of squares pattern is important for several reasons:
- Faster factoring: It allows for quick and efficient factoring of expressions that fit this pattern.
- Simplifying expressions: Factoring expressions can often lead to simplified forms, making further calculations easier.
- Solving equations: The difference of squares pattern is used in solving quadratic equations by factoring.
By understanding and applying the difference of squares pattern, you can effectively manipulate algebraic expressions and simplify mathematical problems.