Choose the correct simplification of the expression .

To choose the correct simplification of the expression \(\frac{14x^3 - 21x^2 - 7x}{-7x}\), we can follow the process of simplifying each term as shown before:

1. **Write the division as a fraction:**
   \[
   \frac{14x^3 - 21x^2 - 7x}{-7x}
   \]

2. **Separate each term in the numerator:**
   \[
   \frac{14x^3}{-7x} - \frac{21x^2}{-7x} - \frac{7x}{-7x}
   \]

3. **Simplify each term:**
   - For the first term: \(\frac{14x^3}{-7x} = -2x^2\)
   - For the second term: \(\frac{21x^2}{-7x} = -3x\)
   - For the third term: \(\frac{7x}{-7x} = -1\)

4. **Combine the simplified terms:**
   \[
   -2x^2 + 3x + 1
   \]

So, the correct simplification of the expression \(\frac{14x^3 - 21x^2 - 7x}{-7x}\) is:
\[
-2x^2 + 3x + 1
\]

Would you like more details or have any questions?

Here are some related questions to further your understanding:

1. How do you perform polynomial long division for more complex expressions?
2. What is the process of synthetic division and when is it used?
3. How do you handle division by a polynomial with more than one term in the denominator?
4. Can you divide polynomials using factorization methods?
5. What are some common mistakes to avoid when dividing polynomials?
6. How do you verify the result of polynomial division?
7. What role does the remainder play in polynomial division?
8. How does polynomial division relate to the concept of finding roots of equations?

**Tip:** Always simplify each term separately when dividing polynomials to avoid mistakes.

Understand how to simplify the expression (14x^3 - 21x^2 - 7x) / (-7x) step-by-step. This problem involves algebraic concepts such as polynomial division and rational expressions, suitable for students in Grades 9-12.

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