Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
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].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
