Questions: What is the simplest form of this rational expression? (4x^2-14x+6)/(x^2-7x+12) (4x-1)/(x-2), where x ≠ 2 and x ≠ 4 (2(2x-1))/(x-4), where x ≠ 3 and x ≠ 4 (2x-1)/(x-2), where x ≠ 2 and x ≠ 3

Transcript text: What is the simplest form of this rational expression? \[ \frac{4 x^{2}-14 x+6}{x^{2}-7 x+12} \] $\frac{4 x-1}{x-2}$, where $x \neq 2$ and $x \neq 4$ $\frac{2(2 x-1)}{x-4}$, where $x \neq 3$ and $x \neq 4$ $\frac{2 x-1}{x-2}$, where $x \neq 2$ and $x \neq 3$

Solution

Solution Steps

Step 1: Factor the Numerator

We start with the numerator $ 4x^2 - 14x + 6 $. By factoring, we find: \[ 4x^2 - 14x + 6 = 2 \left(x - 3\right) \left(2x - 1\right) \]

Step 2: Factor the Denominator

Next, we factor the denominator $ x^2 - 7x + 12 $: \[ x^2 - 7x + 12 = \left(x - 4\right) \left(x - 3\right) \]

Step 3: Simplify the Rational Expression

Now, we can express the original rational expression as: \[ \frac{4x^2 - 14x + 6}{x^2 - 7x + 12} = \frac{2 \left(x - 3\right) \left(2x - 1\right)}{\left(x - 4\right) \left(x - 3\right)} \] We notice that $ (x - 3) $ is a common factor in both the numerator and the denominator, allowing us to simplify: \[ \frac{2(2x - 1)}{x - 4} \]

Step 4: State the Restrictions

It is important to note the restrictions on $ x $ due to the original denominator. The expression is valid for $ x \neq 4 $ and $ x \neq 3 $. Thus, the simplest form of the rational expression is: \[ \frac{2(2x - 1)}{x - 4}, \quad \text{where } x \neq 3 \text{ and } x \neq 4 \]