Factor 64, minus, 27, v, cubed64−27v 
3
  completely.

To factor the expression \( 64 - 27v^3 \), you can recognize this as a **difference of cubes**. The formula for factoring a difference of cubes is:

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

In this case:
- \( 64 = 4^3 \)
- \( 27v^3 = (3v)^3 \)

So, the expression becomes:

\[
64 - 27v^3 = (4^3) - (3v)^3
\]

Now apply the difference of cubes formula:

\[
64 - 27v^3 = (4 - 3v)(4^2 + 4(3v) + (3v)^2)
\]

Simplify the terms inside the second factor:

\[
4^2 = 16, \quad 4(3v) = 12v, \quad (3v)^2 = 9v^2
\]

So, the factorization becomes:

\[
64 - 27v^3 = (4 - 3v)(16 + 12v + 9v^2)
\]

Thus, the completely factored form is:

\[
64 - 27v^3 = (4 - 3v)(16 + 12v + 9v^2)
\]

Would you like further details or have any questions on this?

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Here are 5 questions to expand on the concept:
1. How would you factor a sum of cubes, like \( a^3 + b^3 \)?
2. Can you solve for \( v \) using the factored form?
3. How do you identify if an expression is a perfect cube?
4. What other factoring formulas do you know?
5. How does the difference of cubes compare to the difference of squares?

**Tip:** Recognizing special patterns like the difference of cubes can help you factor complex expressions quickly.

Learn how to factor the polynomial expression 64 - 27v^3 using the difference of cubes formula. This step-by-step guide breaks down the factoring process, including key concepts such as recognizing perfect cubes and applying the difference of cubes theorem. Ideal for students in Grades 9-11.

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