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125t3 - 216u3 = ?

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(5t-6u)(25t2+30tu+36u2)
(5t+6u)(25t2+30tu-36u2)
(5t+6u)(25t2-30tu+36u2)
(5t-6u)(25t2-30tu+36u2)

The given expression is \( 125t^3 - 216u^3 \).

This is a difference of cubes, which can be factored using the formula:

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

Here, \( a = 5t \) and \( b = 6u \), so:

\[
125t^3 - 216u^3 = (5t - 6u)((5t)^2 + (5t)(6u) + (6u)^2)
\]

Expanding the terms inside the second factor:

\[
= (5t - 6u)(25t^2 + 30tu + 36u^2)
\]

Thus, the correct answer is:

**(5t - 6u)(25t^2 + 30tu + 36u^2)**

Would you like details or have any questions?

Here are 8 follow-up questions for further practice:
1. How would you factor \( 64x^3 + 27y^3 \)?
2. What is the general formula for factoring a sum of cubes?
3. Can you explain the difference between a sum of cubes and a difference of cubes?
4. Factor \( 27a^3 - 8b^3 \).
5. How do you expand \( (3x + 4y)^3 \) using the binomial theorem?
6. What is the result of factoring \( 343m^3 - 125n^3 \)?
7. Solve the equation \( 8x^3 + 1 = 0 \).
8. How can you verify the factors of a cubic polynomial?

**Tip:** Always check for special factoring formulas like the difference of squares and cubes to simplify polynomial expressions.

Learn how to factorize the expression 125t^3 - 216u^3 using the difference of cubes formula. This problem involves polynomial factorization and applies the difference of cubes theorem. Suitable for high school students.

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