Round #1046

Simplify the rational expression (x^2 - 9) / (x^2 - x - 6) and state any restrictions on x. Give the simplified form.

## Step 1: Factor the numerator and denominator To simplify the rational expression, we first need to factor both the numerator and the denominator. The numerator is \(x^2 - 9\), which can be factored into \((x + 3)(x - 3)\) since it is a difference of squares. The denominator is \(x^2 - x - 6\), which can be factored into \((x - 3)(x + 2)\) by finding two numbers that multiply to -6 and add up to -1. ## Step 2: Cancel common factors After factoring, the expression becomes \(\frac{(x + 3)(x - 3)}{(x - 3)(x + 2)}\). We can see that \((x - 3)\) is a common factor in both the numerator and the denominator. Therefore, we can cancel out the \((x - 3)\) from both the numerator and the denominator, leaving us with \(\frac{x + 3}{x + 2}\). ## Step 3: State restrictions on x When we canceled \((x - 3)\), we implicitly assumed that \(x \neq 3\) because if \(x = 3\), then both the numerator and the denominator would be zero, which is undefined. Additionally, since the original denominator cannot be zero, we must also consider the restriction from the other factor in the denominator, \(x + 2\), meaning \(x \neq -2\). Thus, the restrictions on \(x\) are \(x \neq 3\) and \(x \neq -2\). The final answer is: $\boxed{\frac{x + 3}{x + 2}}$

To simplify the rational expression (x^2 - 9) / (x^2 - x - 6), we first factor both the numerator and the denominator. The numerator x^2 - 9 can be factored as (x + 3)(x - 3) using the difference of squares formula. The denominator x^2 - x - 6 can be factored as (x - 3)(x + 2) by finding two numbers that multiply to -6 and add up to -1. So, the expression becomes ((x + 3)(x - 3)) / ((x - 3)(x + 2)). We can cancel out the common factor (x - 3) from both the numerator and the denominator, resulting in (x + 3) / (x + 2). However, we must note that x cannot be equal to 3 because that would make the denominator zero before simplification, and division by zero is undefined. Additionally, x cannot be equal to -2 because that would make the denominator zero in the simplified form, which is also undefined. Therefore, the simplified form is (x + 3) / (x + 2), with restrictions that x ≠ 3 and x ≠ -2.