Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's take a look at the expressions [tex]\( P(x) \cdot Q(x) \)[/tex] and [tex]\( P(x) \div Q(x) \)[/tex] and match each one with its simplified form.
Given the functions:
[tex]\[ P(x) = \frac{2}{3x - 1} \][/tex]
[tex]\[ Q(x) = \frac{6}{3x + 2} \][/tex]
### 1. Finding [tex]\( P(x) \cdot Q(x) \)[/tex]
First, we calculate the product [tex]\( P(x) \cdot Q(x) \)[/tex]:
[tex]\[ P(x) \cdot Q(x) = \left(\frac{2}{3x - 1}\right) \cdot \left(\frac{6}{3x + 2}\right) \][/tex]
Multiplying these fractions, we get:
[tex]\[ P(x) \cdot Q(x) = \frac{2 \cdot 6}{(3x - 1)(3x + 2)} = \frac{12}{(3x - 1)(3x + 2)} \][/tex]
So, the simplified form of [tex]\( P(x) \cdot Q(x) \)[/tex] is:
[tex]\[ \frac{12}{(3x - 1)(3x + 2)} \][/tex]
### 2. Finding [tex]\( P(x) \div Q(x) \)[/tex]
Next, we calculate the division [tex]\( P(x) \div Q(x) \)[/tex]:
[tex]\[ P(x) \div Q(x) = \left(\frac{2}{3x - 1}\right) \div \left(\frac{6}{3x + 2}\right) \][/tex]
Dividing fractions is equivalent to multiplying by the reciprocal:
[tex]\[ P(x) \div Q(x) = \frac{2}{3x - 1} \cdot \frac{3x + 2}{6} \][/tex]
Simplifying this, we get:
[tex]\[ P(x) \div Q(x) = \frac{2(3x + 2)}{6(3x - 1)} = \frac{3x + 2}{3(3x - 1)} \][/tex]
So, the simplified form of [tex]\( P(x) \div Q(x) \)[/tex] is:
[tex]\[ \frac{3x + 2}{3(3x - 1)} \][/tex]
### Matching the expressions with their simplified forms
Using the simplified results from above:
- [tex]\( P(x) \cdot Q(x) \longrightarrow \frac{12}{(3x - 1)(3x + 2)} \)[/tex]
- [tex]\( P(x) \div Q(x) \longrightarrow \frac{3x + 2}{3(3x - 1)} \)[/tex]
Given the functions:
[tex]\[ P(x) = \frac{2}{3x - 1} \][/tex]
[tex]\[ Q(x) = \frac{6}{3x + 2} \][/tex]
### 1. Finding [tex]\( P(x) \cdot Q(x) \)[/tex]
First, we calculate the product [tex]\( P(x) \cdot Q(x) \)[/tex]:
[tex]\[ P(x) \cdot Q(x) = \left(\frac{2}{3x - 1}\right) \cdot \left(\frac{6}{3x + 2}\right) \][/tex]
Multiplying these fractions, we get:
[tex]\[ P(x) \cdot Q(x) = \frac{2 \cdot 6}{(3x - 1)(3x + 2)} = \frac{12}{(3x - 1)(3x + 2)} \][/tex]
So, the simplified form of [tex]\( P(x) \cdot Q(x) \)[/tex] is:
[tex]\[ \frac{12}{(3x - 1)(3x + 2)} \][/tex]
### 2. Finding [tex]\( P(x) \div Q(x) \)[/tex]
Next, we calculate the division [tex]\( P(x) \div Q(x) \)[/tex]:
[tex]\[ P(x) \div Q(x) = \left(\frac{2}{3x - 1}\right) \div \left(\frac{6}{3x + 2}\right) \][/tex]
Dividing fractions is equivalent to multiplying by the reciprocal:
[tex]\[ P(x) \div Q(x) = \frac{2}{3x - 1} \cdot \frac{3x + 2}{6} \][/tex]
Simplifying this, we get:
[tex]\[ P(x) \div Q(x) = \frac{2(3x + 2)}{6(3x - 1)} = \frac{3x + 2}{3(3x - 1)} \][/tex]
So, the simplified form of [tex]\( P(x) \div Q(x) \)[/tex] is:
[tex]\[ \frac{3x + 2}{3(3x - 1)} \][/tex]
### Matching the expressions with their simplified forms
Using the simplified results from above:
- [tex]\( P(x) \cdot Q(x) \longrightarrow \frac{12}{(3x - 1)(3x + 2)} \)[/tex]
- [tex]\( P(x) \div Q(x) \longrightarrow \frac{3x + 2}{3(3x - 1)} \)[/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.