To simplify the given product of fractions:
[tex]\[
\frac{x^2 - 3x - 10}{x^2 - 6x + 5} \cdot \frac{x - 2}{x - 5}
\][/tex]
we need to go through the following steps:
1. Factor the expressions in the fractions:
- Factor the numerator [tex]\( x^2 - 3x - 10 \)[/tex]:
[tex]\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\][/tex]
- Factor the denominator [tex]\( x^2 - 6x + 5 \)[/tex]:
[tex]\[
x^2 - 6x + 5 = (x - 1)(x - 5)
\][/tex]
- The second fraction has straightforward expressions [tex]\( \frac{x - 2}{x - 5} \)[/tex].
2. Rewrite the fractions with the factored expressions:
[tex]\[
\frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5}
\][/tex]
3. Multiply the fractions:
- Combine the numerators:
[tex]\[
\text{Numerator} = (x - 5)(x + 2) \cdot (x - 2)
\][/tex]
- Combine the denominators:
[tex]\[
\text{Denominator} = (x - 1)(x - 5) \cdot (x - 5)
\][/tex]
4. Simplify the expressions by canceling common factors:
- The numerator [tex]\( (x - 5)(x + 2)(x - 2) \)[/tex] remains as it is.
- The denominator can be simplified:
[tex]\[
(x - 5)(x - 5)(x - 1) = (x - 5)^2(x - 1)
\][/tex]
5. Simplified products:
The simplest form of the numerator:
[tex]\[
-(x - 2)(-x^2 + 3x + 10)
\][/tex]
The simplest form of the denominator:
[tex]\[
(x - 5)(x^2 - 6x + 5)
\][/tex]
6. Determine the excluded values:
The excluded values are where the denominator equals zero:
[tex]\[
(x - 1)(x - 5) = 0
\][/tex]
[tex]\[
x - 1 = 0 \quad \text{or} \quad x - 5 = 0
\][/tex]
Excluded values:
[tex]\[
x = 1 \quad \text{and} \quad x = 5
\][/tex]
So, the answers are:
- The numerator in its simplest form is [tex]\(-(x - 2)(-x^2 + 3x + 10)\)[/tex].
- The denominator in its simplest form is [tex]\((x - 5)(x^2 - 6x + 5)\)[/tex].
- The excluded values are [tex]\(x = 1\)[/tex] and [tex]\(x = 5\)[/tex].
