To factor the expression \( 12p^3 - 12p \), we start by identifying the greatest common factor (GCF) of the terms.

The terms are \( 12p^3 \) and \( 12p \). The GCF of these terms is \( 12p \).

Next, we factor out the GCF from each term:
\[ 12p^3 - 12p = 12p(p^2 - 1) \]

Now, we notice that \( p^2 - 1 \) is a difference of squares:
\[ p^2 - 1 = (p + 1)(p - 1) \]

So, we can further factor \( p^2 - 1 \) as:
\[ 12p(p^2 - 1) = 12p(p + 1)(p - 1) \]

Thus, the fully factored form of \( 12p^3 - 12p \) is:
\[ 12p(p + 1)(p - 1) \]

Would you like details or have any questions about this factorization?

Here are 5 relative questions to consider:
1. What is the greatest common factor (GCF) and how is it determined?
2. Why is it useful to factor out the GCF first in an expression?
3. What is the difference of squares formula?
4. How can factoring help in solving polynomial equations?
5. Can the process of factoring be applied to polynomials with more than two terms?

**Tip:** Always look for the greatest common factor (GCF) first when factoring an expression, as it simplifies the process and reduces the terms to more manageable components.

Learn how to factor the polynomial 12p^3 - 12p with a detailed step-by-step solution. This problem covers polynomial factorization techniques, including identifying the greatest common factor (GCF) and applying the difference of squares formula. Suitable for high school students studying algebraic factorization.

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