To factor the expression \( 125y^3 - v^3 \), we can recognize this as a difference of cubes. The general formula for factoring a difference of cubes is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, we identify:
- \( a^3 = 125y^3 \) so \( a = 5y \)
- \( b^3 = v^3 \) so \( b = v \)

Using the formula:

\[
125y^3 - v^3 = (5y - v)\left((5y)^2 + (5y)(v) + v^2\right)
\]

Now, simplify the expression:

\[
125y^3 - v^3 = (5y - v)(25y^2 + 5yv + v^2)
\]

So, the factored form is:

\[
\boxed{(5y - v)(25y^2 + 5yv + v^2)}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you factor the sum of cubes, such as \( a^3 + b^3 \)?
2. Can you provide an example of factoring a trinomial like \( 2x^2 + 3x + 1 \)?
3. How would you factor the expression \( 64z^3 - 1 \)?
4. What is the difference between a perfect square trinomial and a difference of cubes?
5. How can you apply factoring to solve polynomial equations?

**Tip:** Recognizing patterns like the difference of cubes helps in factoring expressions quickly.

Learn how to factor the expression 125y^3 - v^3 using the difference of cubes formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring, difference of cubes, and polynomial expressions. Suitable for high school students studying algebra.

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