Which of the following is the quotient of the rational expressions shown below? Make sure your answer is in reduced form.

[tex]\[ \frac{3x - 6}{x^3} \div \frac{x - 2}{2x - 1} \][/tex]

A. [tex][tex]$\frac{6x - 3}{x^3}$[/tex][/tex]
B. [tex][tex]$\frac{3x^2 - 12x + 12}{2x^4 - x^3}$[/tex][/tex]
C. [tex][tex]$\frac{5x - 7}{x^3 + x - 2}$[/tex][/tex]
D. [tex][tex]$\frac{4x - 8}{x^3 + 2x - 1}$[/tex][/tex]
E. [tex][tex]$\frac{6x^2 - 15x + 6}{x^4 - 2x^{\hookrightarrow}}$[/tex][/tex]



Answer :

To solve the problem of finding the quotient of two given rational expressions, let's go through the solution step by step:

First, we are given the following rational expression to divide:
[tex]\[ \frac{3x - 6}{x^3} \div \frac{x - 2}{2x - 1} \][/tex]

1. Rewrite the division as multiplication by the reciprocal:
Division of fractions is equivalent to multiplying by the reciprocal of the second fraction. Therefore, we can rewrite the problem as:
[tex]\[ \frac{3x - 6}{x^3} \times \frac{2x - 1}{x - 2} \][/tex]

2. Multiply the numerators and denominators:
Multiply the numerators together:
[tex]\[ (3x - 6)(2x - 1) \][/tex]
Multiply the denominators together:
[tex]\[ x^3 (x - 2) \][/tex]

3. Combine the expressions:
Putting it all together gives:
[tex]\[ \frac{(3x - 6)(2x - 1)}{x^3 (x - 2)} \][/tex]

4. Simplify the numerator:
The numerator [tex]\((3x - 6)(2x - 1)\)[/tex] simplifies as follows:
[tex]\[ 3x \cdot 2x + 3x \cdot (-1) + (-6) \cdot 2x + (-6) \cdot (-1) = 6x^2 - 3x - 12x + 6 = 6x^2 - 15x + 6 \][/tex]

5. Simplify the denominator:
The denominator [tex]\(x^3 (x - 2)\)[/tex] simplifies as follows:
[tex]\[ x^3 \cdot (x - 2) = x^4 - 2x^3 \][/tex]

6. Combine simplified expressions:
Therefore, the simplified form of the quotient of the rational expressions is:
[tex]\[ \frac{6x^2 - 15x + 6}{x^4 - 2x^3} \][/tex]

Hence, the correct answer among the given options is:

E. [tex]\(\frac{6 x^2 - 15 x + 6}{x^4 - 2 x^3}\)[/tex]