Which of the following is the product of the rational expressions shown below?

[tex]\[ \frac{2}{x} \cdot \frac{3}{2x-5} \][/tex]

A. [tex][tex]$\frac{6}{2x^2-5x}$[/tex][/tex]

B. [tex][tex]$\frac{3}{x^2-5x}$[/tex][/tex]

C. [tex][tex]$\frac{6}{2x-5}$[/tex][/tex]

D. [tex][tex]$\frac{6x}{x-5}$[/tex][/tex]



Answer :

To find the product of the given rational expressions:
[tex]\[ \frac{2}{x} \cdot \frac{3}{2x - 5} \][/tex]
we need to follow these steps:

1. Multiply the numerators:
Identify the numerators of each fraction and multiply them together. In this case, the numerators are 2 and 3.

[tex]\[ 2 \times 3 = 6 \][/tex]

2. Multiply the denominators:
Similarly, identify the denominators of each fraction and multiply them. The denominators are [tex]\( x \)[/tex] and [tex]\( 2x - 5 \)[/tex].

[tex]\[ x \times (2x - 5) = x(2x - 5) \][/tex]

3. Simplify the product of the denominators:
Now, distribute [tex]\( x \)[/tex] in the denominator expression.

[tex]\[ x \times (2x - 5) = 2x^2 - 5x \][/tex]

4. Form the product of the fractions:
Combine the results from the steps above. The product of the rational expressions is the fraction formed by these new numerator and denominator.

[tex]\[ \frac{6}{2x^2 - 5x} \][/tex]

Now, compare this result with the given options:

A. [tex]\(\frac{6}{2x^2 - 5x}\)[/tex] \
B. [tex]\(\frac{3}{x^2 - 5x}\)[/tex] \
C. [tex]\(\frac{6}{2x - 5}\)[/tex] \
D. [tex]\(\frac{6x}{x - 5}\)[/tex]

Clearly, the product we derived matches option A:

[tex]\[ \frac{6}{2x^2 - 5x} \][/tex]

Thus, the correct answer is:

[tex]\[ \boxed{\frac{6}{2x^2 - 5x}} \][/tex]

This corresponds to option A.