Certainly! Let’s break down the given problem step by step:
We need to find the simplest form of the product of the fractions and determine the excluded values for [tex]\( x \)[/tex].
The given product of fractions is:
[tex]\[
\frac{x^2 - 3x - 10}{x^2 - 6x + 5} \cdot \frac{x - 2}{x - 5}
\][/tex]
1. Factor the Numerators and Denominators:
The numerator of the first fraction, [tex]\( x^2 - 3x - 10 \)[/tex], can be factored into:
[tex]\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\][/tex]
The denominator of the first fraction, [tex]\( x^2 - 6x + 5 \)[/tex], can be factored into:
[tex]\[
x^2 - 6x + 5 = (x - 1)(x - 5)
\][/tex]
The numerator of the second fraction is [tex]\( x - 2 \)[/tex] which is already in its simplest form.
The denominator of the second fraction is [tex]\( x - 5 \)[/tex] which is already in its simplest form.
2. Write the Product with Factored Expressions:
[tex]\[
\frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5}
\][/tex]
3. Simplify by Canceling Common Factors:
Cancel [tex]\( x - 5 \)[/tex] from the numerator and denominator (note that [tex]\( x \neq 5 \)[/tex]):
[tex]\[
\frac{(x + 2)}{(x - 1)} \cdot \frac{x - 2}{x - 5}
\][/tex]
This leaves us with:
[tex]\[
\frac{(x + 2)(x - 2)}{(x - 1)(x - 5)}
\][/tex]
4. Multiply the Remaining Expressions:
[tex]\[
\frac{x^2 - 4}{x^2 - 6x + 5}
\][/tex]
Since [tex]\( x^2 - 4 = (x + 2)(x - 2) \)[/tex] (this matches our factored forms previously), the simplest form of the product is indeed:
[tex]\[
\frac{x^2 - 4}{x^2 - 6x + 5}
\][/tex]
5. Determine Excluded Values:
The original problem has factors in the denominator where [tex]\( x - 1 \)[/tex], [tex]\( x - 5 \)[/tex] appear. Thus, these values for [tex]\( x \)[/tex] will make the denominator zero and are excluded from the domain.
- From [tex]\( x - 1 = 0 \Rightarrow x = 1 \)[/tex]
- From [tex]\( x - 5 = 0 \Rightarrow x = 5 \)[/tex]
Therefore, the excluded values of [tex]\( x \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex].
Putting it all together:
The simplest form of the product has:
- Numerator: [tex]\( x^2 - 4 \)[/tex]
- Denominator: [tex]\( x^2 - 6x + 5 \)[/tex]
- Excluded value of [tex]\( x \)[/tex] is [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex]
So, the correct selections are:
The simplest form of this product has a numerator of [tex]\( x^2 - 4 \)[/tex] and a denominator of [tex]\( x^2 - 6x + 5 \)[/tex]. The expression has an excluded value of [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex].
