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Consider the functions given below.

[tex]\[
\begin{array}{l}
P(x)=\frac{2}{3 x-1} \\
Q(x)=\frac{6}{3 x+2}
\end{array}
\][/tex]

Match each expression with its simplified form.

[tex]\[
\begin{array}{l}
\begin{array}{cc}
\frac{3(3 x-1)}{-3 x+2} & \frac{2(12 x+1)}{(3 x-1)(-3 x+2)} \\
\frac{-2(12 x-5)}{3 x-1)(-3 x+2)} & \frac{12}{(3 x-1)(-3 x+2)}
\end{array} \\
\frac{2(6 x-1)}{(3 x-1)(-3 x+2)} \\
\frac{-3 x+2}{3(3 x-1)} \\
\frac{-2(12 x-5)}{(3 x-1)(-3 x+2)} \frac{12}{(3 x-1)(-3 x+2)} \\
P(x) \cdot Q(x) \longrightarrow \\
P(x) \div Q(x) \longrightarrow \\
\end{array}
\][/tex]

Sagot :

GinnyAnswer
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]
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