%2F(x%2B3).jpeg "(x^2+8x+15)/(x+3)")
Simplifying the Expression (x^2 + 8x + 15) / (x + 3)
This expression represents a rational function, where the numerator is a quadratic polynomial and the denominator is a linear polynomial. We can simplify this expression by factoring the numerator and canceling common factors.
Factoring the Numerator
The numerator, x^2 + 8x + 15, is a quadratic expression that can be factored into two binomials. We need to find two numbers that add up to 8 (the coefficient of the x term) and multiply to 15 (the constant term). These numbers are 3 and 5:
- 3 + 5 = 8
- 3 * 5 = 15
Therefore, we can factor the numerator as:
(x + 3)(x + 5)
Simplifying the Expression
Now, we can rewrite the original expression with the factored numerator:
(x^2 + 8x + 15) / (x + 3) = (x + 3)(x + 5) / (x + 3)
Notice that the factor (x + 3) appears in both the numerator and the denominator. We can cancel out these common factors:
(x + 3)(x + 5) / (x + 3) = x + 5
Conclusion
The simplified form of the expression (x^2 + 8x + 15) / (x + 3) is x + 5, with the restriction that x ≠ -3. This restriction is necessary because the original expression is undefined when x = -3, as it would lead to division by zero.