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]
