(a-3).jpeg "(a+3)(a-3)")
Understanding the Expansion of (a + 3)(a - 3)
The expression (a + 3)(a - 3) represents the product of two binomials. To understand its expansion, we can utilize the FOIL method, which stands for First, Outer, Inner, Last.
Expanding Using FOIL Method
- First: Multiply the first terms of each binomial: a * a = a²
- Outer: Multiply the outer terms of the binomials: a * -3 = -3a
- Inner: Multiply the inner terms of the binomials: 3 * a = 3a
- Last: Multiply the last terms of each binomial: 3 * -3 = -9
Now, we combine all the terms: a² - 3a + 3a - 9
Finally, we simplify by combining the like terms: a² - 9
The Difference of Squares Pattern
The expansion of (a + 3)(a - 3) results in a² - 9. This is a classic example of the difference of squares pattern.
The Difference of Squares Pattern: (x + y)(x - y) = x² - y²
In our case, x = a and y = 3.
Significance of the Difference of Squares
The difference of squares pattern is a valuable tool for factoring and simplifying algebraic expressions. It helps us recognize and manipulate expressions that follow this specific pattern.
Example:
If we encounter an expression like a² - 16, we can recognize it as the difference of squares (a² - 4²). Therefore, we can factor it as (a + 4)(a - 4).
Conclusion
The expansion of (a + 3)(a - 3) demonstrates the application of the FOIL method and the difference of squares pattern. By understanding these concepts, we can effectively simplify and manipulate algebraic expressions, making calculations and problem-solving more efficient.