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Understanding the Expansion of (x-b)(x+b)
The expression (x-b)(x+b) is a common algebraic pattern known as the difference of squares. This pattern allows us to simplify the multiplication of two binomials in a quick and efficient manner.
Expanding the Expression
To expand the expression (x-b)(x+b), we can use the FOIL method:
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms of the binomials: x * b = xb
- Inner: Multiply the inner terms of the binomials: -b * x = -xb
- Last: Multiply the last terms of each binomial: -b * b = -b²
Combining these terms, we get:
x² + xb - xb - b²
Notice that the terms xb and -xb cancel each other out, leaving us with:
x² - b²
The Difference of Squares Pattern
The result, x² - b², demonstrates the difference of squares pattern:
(x - b)(x + b) = x² - b²
This pattern highlights that the product of two binomials, where one is the sum of two terms and the other is their difference, always results in the difference of the squares of those terms.
Applications of the Difference of Squares
The difference of squares pattern is a useful tool in various algebraic situations, including:
- Factoring expressions: The pattern can be used to factor expressions of the form x² - b² into (x - b)(x + b).
- Simplifying equations: Recognizing the pattern can help simplify equations involving the difference of squares.
- Solving equations: The pattern can be used to solve equations involving the difference of squares by factoring and setting each factor to zero.
Examples
Here are some examples of how the difference of squares pattern can be applied:
1. Factoring:
Factor the expression x² - 9:
x² - 9 = (x - 3)(x + 3)
2. Simplifying:
Simplify the expression (x + 5)(x - 5):
(x + 5)(x - 5) = x² - 25
3. Solving:
Solve the equation x² - 16 = 0:
x² - 16 = (x - 4)(x + 4) = 0
Therefore, x = 4 or x = -4.
Conclusion
The difference of squares pattern is a powerful tool for simplifying algebraic expressions and solving equations. Understanding this pattern can enhance your algebraic skills and make problem-solving more efficient.