To factor the expression \( 27w^3 - 125 \), we can recognize this as a **difference of cubes**. The general formula for factoring a difference of cubes is:

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

In this case:
- \( a^3 = 27w^3 \), so \( a = 3w \)
- \( b^3 = 125 \), so \( b = 5 \)

Now applying the formula:

\[
27w^3 - 125 = (3w - 5)\left((3w)^2 + (3w)(5) + (5)^2\right)
\]

Simplify:

\[
= (3w - 5)(9w^2 + 15w + 25)
\]

Thus, the factored form is:

\[
27w^3 - 125 = (3w - 5)(9w^2 + 15w + 25)
\]

Would you like any additional details or have any questions?

Here are 5 related questions:
1. How would you factor the expression \( 64x^3 - 8y^3 \)?
2. Can you explain how to factor a sum of cubes?
3. What are the steps to simplify \( (2x - 3)(4x^2 + 6x + 9) \)?
4. How do you factor higher-degree polynomials?
5. What is the difference between factoring a quadratic and a cubic expression?

**Tip:** Always check if a polynomial can be factored using special formulas like the difference of cubes or sum of cubes to simplify your work!

Learn how to factor the polynomial 27w^3 - 125 by applying the difference of cubes formula. This problem demonstrates how to simplify cubic polynomials and is suitable for students in Grades 9-12.

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