Let's find the simplified forms for [tex]\( P(x) \div Q(x) \)[/tex] and [tex]\( P(x) \cdot Q(x) \)[/tex] by following step-by-step algebraic operations.
First, let's handle [tex]\( P(x) \div Q(x) \)[/tex].
Given:
[tex]\[ P(x) = \frac{2}{3x-1} \][/tex]
[tex]\[ Q(x) = \frac{6}{3x+2} \][/tex]
For division:
[tex]\[ P(x) \div Q(x) = \frac{\frac{2}{3x-1}}{\frac{6}{3x+2}} \][/tex]
To simplify, multiply by the reciprocal of [tex]\( Q(x) \)[/tex]:
[tex]\[ P(x) \div Q(x) = \frac{2}{3x-1} \times \frac{3x+2}{6} \][/tex]
Simplify the numerator and denominator:
[tex]\[ P(x) \div Q(x) = \frac{2(3x+2)}{6(3x-1)} \][/tex]
Factor as much as possible:
[tex]\[ P(x) \div Q(x) = \frac{2(3x+2)}{6(3x-1)} = \frac{(3x+2)}{3(3x-1)} \][/tex]
So, for [tex]\( P(x) \div Q(x) \)[/tex]:
[tex]\[ P(x) \div Q(x) \longrightarrow \frac{3x+2}{3(3x-1)} \][/tex]
Now, let's handle [tex]\( P(x) \cdot Q(x) \)[/tex].
For multiplication:
[tex]\[ P(x) \cdot Q(x) = \frac{2}{3x-1} \times \frac{6}{3x+2} \][/tex]
Multiply the numerators and the denominators:
[tex]\[ P(x) \cdot Q(x) = \frac{2 \times 6}{(3x-1) \times (3x+2)} \][/tex]
Simplify the multiplication:
[tex]\[ P(x) \cdot Q(x) = \frac{12}{(3x-1)(3x+2)} \][/tex]
So, for [tex]\( P(x) \cdot Q(x) \)[/tex]:
[tex]\[ P(x) \cdot Q(x) \longrightarrow \frac{12}{(3x-1)(3x+2)} \][/tex]
Thus, the correct matching pairs are:
[tex]\[ P(x) \div Q(x) \longrightarrow \frac{3x+2}{3(3x-1)} \][/tex]
[tex]\[ P(x) \cdot Q(x) \longrightarrow \frac{12}{(3x-1)(3x+2)} \][/tex]
