The expression you've provided is:

\[
\frac{2x^3}{(x-1)(x-2)(x-3)}
\]

This is a rational function. Here's how it can be analyzed or simplified:

1. **Domain:** The function is undefined where the denominator equals zero. So, \(x \neq 1\), \(x \neq 2\), and \(x \neq 3\).

2. **Factorization:** The numerator is already a simple polynomial \(2x^3\), and the denominator is a product of three linear factors \((x - 1)(x - 2)(x - 3)\).

3. **Long Division or Partial Fraction Decomposition:** To simplify or break down this function, we might use long division (if the degrees of numerator and denominator are appropriate) or partial fraction decomposition to express it as a sum of simpler fractions.

Would you like to simplify this expression further using partial fractions or another method? Or do you have a specific question related to this expression?

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Here are 8 follow-up questions that might interest you:

1. What are partial fractions, and how can they be used to simplify rational expressions?
2. How do you perform long division on polynomials?
3. What are the roots of the numerator, and how do they affect the function's graph?
4. How would you find the asymptotes of this rational function?
5. What is the importance of the domain in rational functions?
6. Can you analyze the behavior of this function as \(x\) approaches 1, 2, or 3?
7. How would you graph this function?
8. What are the key characteristics of rational functions?

**Tip:** Understanding partial fractions can be extremely helpful in calculus, especially when integrating rational functions.

Explore how to simplify the rational function 2x^3 / ((x-1)(x-2)(x-3)) step-by-step. Understand the domain restrictions and methods like partial fraction decomposition. Suitable for high school students studying algebra and rational functions.

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