(a-3)(a2%2B9).jpeg "(a+3)(a-3)(a2+9)")
Factoring the Expression (a+3)(a-3)(a^2+9)
This expression is a classic example of a product of difference of squares and sum of squares. Let's break down the factorization step-by-step:
Difference of Squares Pattern
- The first two factors, (a+3) and (a-3), represent the difference of squares. This is because they follow the form (x + y)(x - y) where x = a and y = 3.
- Using the difference of squares formula: (x + y)(x - y) = x^2 - y^2
- We can simplify the first two factors as follows:
- (a + 3)(a - 3) = a^2 - 3^2 = a^2 - 9
Sum of Squares Pattern
- The third factor, (a^2 + 9), represents the sum of squares.
- Importantly, there is no direct formula for factoring the sum of squares over real numbers.
Final Factorization
- Combining the simplified first two factors with the third factor, we get:
- (a^2 - 9)(a^2 + 9)
- This is the fully factored form of the expression.
Conclusion
Therefore, the fully factored form of the expression (a+3)(a-3)(a^2+9) is (a^2 - 9)(a^2 + 9). It's important to note that the sum of squares term (a^2 + 9) cannot be factored further over the real numbers.