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Let's analyze and solve the given expression.

The expression is:

\[
\frac{(8y^7 z^2 - 19y^7 z^7)}{-2y^3 z^4}
\]

### Step 1: Factor the numerator
First, we factor out the common terms in the numerator:

\[
8y^7 z^2 - 19y^7 z^7 = y^7 (8z^2 - 19z^7)
\]

### Step 2: Rewrite the expression
Now the expression looks like:

\[
\frac{y^7 (8z^2 - 19z^7)}{-2y^3 z^4}
\]

### Step 3: Simplify the expression
Next, we simplify the powers of \(y\) and \(z\):

- For \(y\), we have \(y^7\) in the numerator and \(y^3\) in the denominator. Subtract the exponents:
\[
y^{7-3} = y^4
\]

- For \(z\), we can divide the entire expression by \(z^4\), leaving:
\[
\frac{8z^2}{z^4} = 8z^{-2} = \frac{8}{z^2}
\]
and
\[
\frac{-19z^7}{z^4} = -19z^{7-4} = -19z^3
\]

Thus, the expression becomes:
\[
\frac{y^4 (8z^{-2} - 19z^3)}{-2}
\]

### Step 4: Simplify further
We can now divide the entire expression by \(-2\):

\[
\frac{y^4 (8z^{-2} - 19z^3)}{-2} = -\frac{y^4}{2} (8z^{-2} - 19z^3)
\]

This is the simplified expression.

---

Would you like more details on any specific step? Here are five related questions to consider:

1. How do you factor out common terms from polynomials?
2. Why do we subtract exponents when dividing like bases?
3. How do we simplify negative exponents like \(z^{-2}\)?
4. What happens when dividing a polynomial by a constant?
5. How can this expression be further simplified if we need an explicit numerical evaluation?

**Tip:** When dividing variables with exponents, remember to subtract the exponents if the base is the same!

Divide and simplify: (8y^7 z^2 - 19y^7 z^7) ÷ (-2y^3 z^4)

Learn how to simplify the division of polynomials involving exponents: (8y^7 z^2 - 19y^7 z^7) ÷ (-2y^3 z^4). This solution covers exponent rules, polynomial factoring, and simplification. Ideal for students in Grades 9-11.

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