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Let's solve the division of the polynomial by the monomial step by step.

The given expression is:

\[
\frac{13z^6y^7 - 12z^7y - 6z^6y^4}{-2z^4y^3}
\]

We will divide each term of the polynomial in the numerator by the monomial in the denominator.

### Step 1: Divide \( 13z^6y^7 \) by \( -2z^4y^3 \)

\[
\frac{13z^6y^7}{-2z^4y^3} = \frac{13}{-2}z^{6-4}y^{7-3} = -\frac{13}{2}z^2y^4
\]

### Step 2: Divide \( -12z^7y \) by \( -2z^4y^3 \)

\[
\frac{-12z^7y}{-2z^4y^3} = \frac{-12}{-2}z^{7-4}y^{1-3} = 6z^3y^{-2} = 6z^3 \cdot \frac{1}{y^2} = \frac{6z^3}{y^2}
\]

### Step 3: Divide \( -6z^6y^4 \) by \( -2z^4y^3 \)

\[
\frac{-6z^6y^4}{-2z^4y^3} = \frac{-6}{-2}z^{6-4}y^{4-3} = 3z^2y
\]

### Final Expression:

Now, combining all terms:

\[
-\frac{13}{2}z^2y^4 + \frac{6z^3}{y^2} + 3z^2y
\]

This is the simplified result.

Would you like any further details or have any questions?

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Here are five questions to expand your understanding:
1. What rules govern the division of exponents when dividing terms with the same base?
2. How do negative exponents affect the outcome of division?
3. How would the result change if a different variable was introduced?
4. Can you rewrite the expression with a common denominator for better clarity?
5. How do you handle negative signs in algebraic expressions involving fractions?

**Tip:** When dividing terms with exponents, subtract the exponent in the denominator from the exponent in the numerator to simplify the expression.

Divide (13z^6y^7 - 12z^7y - 6z^6y^4) ÷ (-2z^4y^3). Simplify your answer as much as possible.

Learn how to divide a multivariate polynomial by a monomial with the problem (13z^6y^7 - 12z^7y - 6z^6y^4) ÷ (-2z^4y^3). This solution involves using exponent rules and simplifying each term step by step. Suitable for students in Grades 8-10.

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