Question content area top
Part 1
Divide.
StartFraction x squared minus 25 Over 6 x EndFraction divided by StartFraction 5 minus x Over 3 xy EndFraction
Question content area bottom
Part 1
StartFraction x squared minus 25 Over 6 x EndFraction divided by StartFraction 5 minus x Over 3 xy EndFraction
equals
  
enter your response here ​(Simplify your​ answer.)

We are asked to divide two rational expressions:

\[
\frac{x^2 - 25}{6x} \div \frac{5 - x}{3xy}
\]

### Step 1: Rewrite division as multiplication
When dividing by a fraction, we multiply by its reciprocal:

\[
\frac{x^2 - 25}{6x} \times \frac{3xy}{5 - x}
\]

### Step 2: Factor where possible
We can factor the numerator \(x^2 - 25\), which is a difference of squares:

\[
x^2 - 25 = (x - 5)(x + 5)
\]

So the expression becomes:

\[
\frac{(x - 5)(x + 5)}{6x} \times \frac{3xy}{5 - x}
\]

### Step 3: Simplify the expression
Notice that \(5 - x = -(x - 5)\). So we can replace \(5 - x\) with \(-(x - 5)\):

\[
\frac{(x - 5)(x + 5)}{6x} \times \frac{3xy}{-(x - 5)}
\]

Now, we can cancel out the common factor of \((x - 5)\):

\[
\frac{(x + 5)}{6x} \times \frac{3xy}{-1}
\]

### Step 4: Multiply the fractions
Now, multiply the numerators and denominators:

\[
\frac{(x + 5)(3xy)}{6x(-1)}
\]

This simplifies to:

\[
\frac{-3xy(x + 5)}{6x}
\]

### Step 5: Cancel common factors
We can cancel \(x\) from the numerator and denominator:

\[
\frac{-3y(x + 5)}{6}
\]

Finally, simplify the constants:

\[
\frac{-y(x + 5)}{2}
\]

### Final Answer:
\[
\frac{-y(x + 5)}{2}
\]

Let me know if you'd like any clarifications!

---
Here are 5 related questions:
1. How would the solution change if the expression involved a difference instead of a product?
2. Can you explain how to factor higher-degree polynomials similarly?
3. What would happen if there were additional variables in the expression?
4. How do you handle division of rational expressions with exponents greater than 2?
5. What other types of factoring could be used to simplify more complex terms?

**Tip**: Always look for common factors in both the numerator and denominator when simplifying rational expressions!

Divide: (x^2 - 25) / (6x) divided by (5 - x) / (3xy). Simplify your answer.

Learn how to simplify rational expressions through division and factoring. This step-by-step solution simplifies (x^2 - 25) / (6x) ÷ (5 - x) / (3xy) using algebraic methods like factoring and reciprocal property. Suitable for Grades 9-11 students.

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