What is the simplified form of the rational expression:
4x−12

3x2−6x−9
?

Question

What is the simplified form of the rational expression: 
4x−12

3x2−6x−9
 ?

BlackTom AI · Accepted Answer

The task presents a rational expression to simplify: (4x − 12) / (3x^2 − 6x − 9). 
First, factor both numerator and denominator where possible. The numerator 4x − 12 factors as 4(x − 3). 
The denominator 3x^2 − 6x − 9 factors by pulling out a 3: 3(x^2 − 2x − 3). Then factor the quadratic x^2 − 2x − 3 as (x − 3)(x + 1). So the denominator becomes 3(x − 3)(x + 1).
Now the original fraction is [4(x − 3)] / [3(x − 3)(x + 1)]. Since the factor (x − 3) appears in both numerator and denominator, it can be canceled, provided we note the domain restriction x ≠ 3 (to avoid division by zero before cancellation).
After cancellation, the expression simplifies to 4 / [3(x + 1)], i.e., 4 divided by (3x + 3). 
In terms of domain, the original expression is undefined at any x that makes the denominator zero, which occurs when x − 3 = 0 or x + 1 = 0, i.e., x = 3 or x = −1. After simplification, the restricted values still include x = 3 (the cancellation point) and x = −1 (where the simplified denominator would be zero).
Since there are no given answer choices in the prompt to evaluate, we cannot analyze specific incorrect options. However, the key steps—factoring, canceling common factors, and noting domain restrictions—are essential to verify any proposed simplified form of this expression.

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