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The given problem is asking to divide a polynomial by a monomial. The expression provided is:

\[
\frac{6u^5 + 19u^4 + 15u^3}{3u^3}
\]

### Step-by-Step Solution:

1. **Separate the terms**: 
   Each term in the numerator should be divided individually by \( 3u^3 \).

\[
\frac{6u^5}{3u^3} + \frac{19u^4}{3u^3} + \frac{15u^3}{3u^3}
\]

2. **Simplify each term**:
   - First term: 
     \[
     \frac{6u^5}{3u^3} = 2u^{5-3} = 2u^2
     \]
   - Second term:
     \[
     \frac{19u^4}{3u^3} = \frac{19}{3}u^{4-3} = \frac{19}{3}u
     \]
   - Third term:
     \[
     \frac{15u^3}{3u^3} = 5u^{3-3} = 5
     \]

3. **Combine the simplified terms**:
   \[
   2u^2 + \frac{19}{3}u + 5
   \]

Thus, the simplified expression is:

\[
2u^2 + \frac{19}{3}u + 5
\]

Would you like further details or have any questions?

### Related Questions:
1. How would you simplify if the exponent in the denominator was different?
2. Can you explain how to handle division when dealing with negative exponents?
3. What happens if one of the coefficients in the numerator was zero?
4. How would you divide by a more complex polynomial, instead of a monomial?
5. Can you generalize the rule for dividing polynomials by monomials?

### Tip:
When dividing a polynomial by a monomial, always subtract the exponents of the variable and reduce the coefficients.

Divide: (6u^5 + 19u^4 + 15u^3) / (3u^3). Simplify your answer as much as possible.

Learn how to divide the polynomial (6u^5 + 19u^4 + 15u^3) by the monomial 3u^3. This problem covers key algebraic concepts, including polynomial division and simplifying exponents. Suitable for students in Grades 8-10.

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