The expression you provided is:

\[
p^5 - 81p
\]

To simplify or factor this expression, we can factor out the greatest common factor (GCF), which is \( p \).

\[
p^5 - 81p = p(p^4 - 81)
\]

Now, we can see if we can factor the expression \( p^4 - 81 \) further. Notice that \( p^4 - 81 \) is a **difference of squares**, because:

\[
p^4 - 81 = (p^2)^2 - 9^2
\]

Using the difference of squares formula:

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

We apply this to \( p^4 - 81 \):

\[
p^4 - 81 = (p^2 - 9)(p^2 + 9)
\]

Now, the expression is factored as:

\[
p(p^2 - 9)(p^2 + 9)
\]

Finally, notice that \( p^2 - 9 \) is also a difference of squares:

\[
p^2 - 9 = (p - 3)(p + 3)
\]

So, the fully factored form of the original expression is:

\[
p(p - 3)(p + 3)(p^2 + 9)
\]

Would you like more details or explanations on any part?

Here are 5 related questions to explore:

1. How do you factor a difference of squares?
2. Can \( p^2 + 9 \) be factored further?
3. How do you find the greatest common factor (GCF) in expressions?
4. What other types of factoring techniques are there in algebra?
5. Can we apply factoring techniques to solve equations?

**Tip**: Always check if you can factor an expression by looking for common factors and special patterns like the difference of squares.

Learn how to factor the algebraic expression p^5 - 81p by applying the greatest common factor (GCF) and the difference of squares. This detailed solution breaks down the steps, highlighting key algebraic techniques like factoring and recognizing special patterns. Suitable for students in Grades 9-12.

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