To simplify the expression (x² + 12x + 35) / (3x + 15), we will follow these steps:
- Factor the Numerator: We need to factor the quadratic expression in the numerator:
- Look for two numbers that multiply to the constant term, 35, and add up to the linear coefficient, 12. The numbers 5 and 7 work because:
- 5 * 7 = 35
- 5 + 7 = 12
- Thus, we can rewrite:
- x² + 12x + 35 = (x + 5)(x + 7)
- Factor the Denominator: Next, let’s factor the denominator:
- The expression 3x + 15 can be factored by taking out the common factor of 3:
- 3x + 15 = 3(x + 5)
- Rewrite the Expression: Now we can rewrite the original expression with the factored forms:
- Our expression now looks like:
- (x + 5)(x + 7) / 3(x + 5)
- Simplify the Expression: We notice that (x + 5) is a common factor in both the numerator and the denominator:
- As long as x ≠ -5 (since we cannot divide by zero), we can cancel (x + 5):
- This leaves us with:
- (x + 7) / 3
Final Result: Therefore, the simplified form of the expression (x² + 12x + 35) / (3x + 15) is:
(x + 7) / 3