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]
