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To factor the expression \( q^3 r^3 - 125 \), we can recognize it as a difference of two cubes.

The difference of two cubes formula is:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

### Step 1: Rewrite Each Term as a Cube
In this expression:
- \( q^3 r^3 \) is already in the form of a cube, specifically \( (qr)^3 \).
- \( 125 \) can be written as \( 5^3 \).

So, we can rewrite the expression as:
\[
(qr)^3 - 5^3
\]

### Step 2: Apply the Difference of Cubes Formula
Now, using \( a = qr \) and \( b = 5 \), we can substitute into the difference of cubes formula:
\[
(qr)^3 - 5^3 = (qr - 5)((qr)^2 + (qr)(5) + 5^2)
\]

### Step 3: Simplify Each Term
1. The first factor is \( qr - 5 \).
2. For the second factor:
   - \((qr)^2 = q^2 r^2\)
   - \((qr)(5) = 5qr\)
   - \(5^2 = 25\)

So, the expression becomes:
\[
(qr - 5)(q^2 r^2 + 5qr + 25)
\]

### Final Answer
The factored form of \( q^3 r^3 - 125 \) is:
\[
(qr - 5)(q^2 r^2 + 5qr + 25)
\]

Would you like further details on the steps or have any questions?

### Related Questions:
1. How can you factor the sum of two cubes?
2. How do you recognize an expression as a difference of cubes?
3. What is the general process for factoring polynomial expressions?
4. Can you apply this formula to complex numbers?
5. How would the solution change if the expression were \( q^3 r^3 + 125 \)?

### Tip:
Remember, for difference and sum of cubes, having the formula memorized will help you factor quickly: \((a - b)(a^2 + ab + b^2)\) for a difference and \((a + b)(a^2 - ab + b^2)\) for a sum.

Factor the following difference of two cubes: q^3 r^3 - 125

This page explains how to factor the expression q^3 r^3 - 125 by applying the difference of cubes formula. Suitable for high school students in Grades 9-12, this guide provides step-by-step instructions and simplifies the process.

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