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Understanding (x+10)(x-10)
The expression (x+10)(x-10) represents a product of two binomials. This type of multiplication is commonly referred to as "FOIL", which stands for First, Outer, Inner, Last.
Expanding the Expression
Let's apply the FOIL method to expand the expression:
- First: Multiply the first terms of each binomial: (x)(x) = x²
- Outer: Multiply the outer terms: (x)(-10) = -10x
- Inner: Multiply the inner terms: (10)(x) = 10x
- Last: Multiply the last terms: (10)(-10) = -100
Now, combining the terms, we have:
x² - 10x + 10x - 100
Simplifying the expression, we get:
x² - 100
Recognizing a Special Pattern
The expression (x+10)(x-10) is a specific example of a difference of squares. This pattern applies when we have two binomials where the only difference is the sign between the terms.
General Pattern: (a + b)(a - b) = a² - b²
In our example, a = x and b = 10. Therefore, the expanded form (x² - 100) is the difference of squares.
Applications
Understanding the difference of squares pattern is useful for:
- Factoring expressions: You can quickly factor an expression like x² - 100 by recognizing it as a difference of squares.
- Solving equations: If you have an equation like x² - 100 = 0, you can easily factor it into (x + 10)(x - 10) = 0, making it easier to solve.
- Simplifying algebraic expressions: By recognizing the difference of squares pattern, you can simplify complex expressions involving these types of products.
In conclusion, the expression (x+10)(x-10) expands to x² - 100, which is a classic example of the difference of squares pattern. Understanding this pattern is crucial for efficient algebraic manipulation and problem-solving.