(x-3)(x%2B7).jpeg "(x+3)(x-3)(x+7)")
Factoring and Expanding (x+3)(x-3)(x+7)
This expression involves multiplying three binomials: (x+3), (x-3), and (x+7). We can solve it through two methods:
1. Expanding the expression
Step 1: Expand the first two binomials, (x+3) and (x-3). This is a special case of the "difference of squares" pattern: (a+b)(a-b) = a² - b²
- (x+3)(x-3) = x² - 3² = x² - 9
Step 2: Multiply the result (x² - 9) by the remaining binomial (x+7)
- (x² - 9)(x + 7) = x²(x + 7) - 9(x + 7)
- = x³ + 7x² - 9x - 63
Therefore, the expanded form of (x+3)(x-3)(x+7) is x³ + 7x² - 9x - 63
2. Using the distributive property
Step 1: Multiply the first two binomials (x+3) and (x-3) using the distributive property.
- (x+3)(x-3) = x(x-3) + 3(x-3)
- = x² - 3x + 3x - 9
- = x² - 9
Step 2: Multiply the result (x² - 9) by the remaining binomial (x+7), again using the distributive property.
- (x² - 9)(x + 7) = x²(x + 7) - 9(x + 7)
- = x³ + 7x² - 9x - 63
Therefore, the expanded form of (x+3)(x-3)(x+7) is x³ + 7x² - 9x - 63
Conclusion
Both methods, expanding and distributive property, lead to the same result: x³ + 7x² - 9x - 63. The choice of method depends on personal preference and the specific problem at hand.