To simplify the expression:

\[
\frac{3x^3 + x^2 + 2x}{9x^3}
\]

We can break this down by dividing each term in the numerator by the denominator \( 9x^3 \).

### Step-by-Step Simplification:
1. **First term:**
   \[
   \frac{3x^3}{9x^3} = \frac{3}{9} = \frac{1}{3}
   \]

2. **Second term:**
   \[
   \frac{x^2}{9x^3} = \frac{1}{9x}
   \]

3. **Third term:**
   \[
   \frac{2x}{9x^3} = \frac{2}{9x^2}
   \]

### Final Simplified Expression:
\[
\frac{1}{3} + \frac{1}{9x} + \frac{2}{9x^2}
\]

Would you like more details or have any questions about this simplification?

Here are 5 related questions to expand on the topic:
1. How would you simplify a similar expression with higher powers of \( x \)?
2. Can you apply this method to simplify fractions with polynomials in both numerator and denominator?
3. How does dividing powers of \( x \) affect the result in general?
4. What happens if we add constants to the denominator before simplifying?
5. Can you factor the numerator in any situation to simplify the expression further?

**Tip:** Always check if terms in the numerator can be factored before dividing by the denominator. This can sometimes lead to a faster simplification.

Learn how to simplify the rational expression 3x^3 + x^2 + 2x / 9x^3 with a clear step-by-step guide. This problem covers key algebraic concepts such as polynomial division, the laws of exponents, and simplifying rational expressions. Ideal for students in Grades 9-12.

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