The expression \( p^5 \cdot (1 - p)^5 \) can be simplified as follows:

1. Notice that both terms are raised to the same power. Therefore, we can rewrite the product as a single power:

\[
p^5 \cdot (1 - p)^5 = \left(p \cdot (1 - p)\right)^5
\]

2. The expression now becomes:

\[
\left(p(1 - p)\right)^5
\]

This is the simplified form, though the inner term \( p(1 - p) \) is already simplified further as a quadratic expression.

If you need further expansion or a specific result, let me know!

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Here are some related questions for further practice:
1. What is the derivative of \( p^5 \cdot (1 - p)^5 \) with respect to \( p \)?
2. How do you expand \( \left(p(1 - p)\right)^5 \) using the binomial theorem?
3. What is the maximum value of the expression \( p^5 \cdot (1 - p)^5 \) for \( p \in [0, 1] \)?
4. What is the second derivative of \( \left(p(1 - p)\right)^5 \) with respect to \( p \)?
5. How do you solve \( p^5 \cdot (1 - p)^5 = 0 \) for \( p \)?

**Tip:** Whenever you see terms raised to the same power, consider factoring the common base or using binomial expansions to simplify!

Discover how to simplify the expression p^5 * (1 - p)^5 by factoring and rewriting powers of binomials. Learn essential algebraic techniques, including simplification of polynomials. This step-by-step solution is suitable for high school students (Grades 9-12).

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