To divide the polynomial \( 14x^3 - 21x^2 - 7x \) by \(-7x\), follow these steps:

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 result of dividing \( 14x^3 - 21x^2 - 7x \) by \(-7x\) 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 check if the polynomial can be factored before dividing, as factorization can simplify the process and help avoid errors.

Learn how to divide the polynomial 14x^3 - 21x^2 - 7x by -7x with a detailed step-by-step solution. This problem covers polynomial division and linear algebra concepts. Suitable for students in Grades 10-12.

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