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Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.

Consider the functions given below.

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

Match each expression with its simplified form.

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

Sagot :

GinnyAnsweridn
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]
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