Math Snap

STEP 1

Assumptions1. We are given two rational expressions $\frac{3 x-9}{x^{}-6 x+9}$ and $\frac{4 x-8}{x^{}-4 x+4}$.
. We need to simplify these expressions and find the numbers that must be excluded from the domain.

STEP 2

Let's start with the first expression. We can simplify the numerator and the denominator by factoring.
$$\frac{ x-9}{x^{2}-6 x+9} = \frac{(x-)}{(x-)^2}$$

STEP 3

Now, we can simplify the expression by cancelling out the common factor $(x-3)$ from the numerator and the denominator.
$$\frac{3(x-3)}{(x-3)^2} = \frac{3}{x-3}$$

STEP 4

The denominator of a fraction cannot be zero, so we need to find the values of $x$ that make the denominator zero.$$x-3 =0$$

STEP 5

olving the equation $x-3 =0$ gives us the value of $x$ that must be excluded from the domain.
$$x =3$$

STEP 6

Now, let's move on to the second expression. We can simplify the numerator and the denominator by factoring.
$$\frac{4 x-8}{x^{2}-4 x+4} = \frac{4(x-2)}{(x-2)^2}$$

STEP 7

Now, we can simplify the expression by cancelling out the common factor $(x-2)$ from the numerator and the denominator.
$$\frac{4(x-2)}{(x-2)^2} = \frac{4}{x-2}$$

STEP 8

The denominator of a fraction cannot be zero, so we need to find the values of $x$ that make the denominator zero.$$x-2 =0$$

olving the equation $x-2 =$ gives us the value of $x$ that must be excluded from the domain.
$$x =2$$So, the simplified expressions are $\frac{3}{x-3}$ and $\frac{4}{x-2}$, and the numbers that must be excluded from the domain are $3$ and $2$ respectively.