Factorize

• p⁴ - 81

To factor the expression \( p^4 - 81 \), we can recognize that it is a difference of squares. The general formula for factoring a difference of squares is \( a^2 - b^2 = (a - b)(a + b) \).

Identifying the Squares

In this case, we can rewrite \( p^4 - 81 \) as:

• \( p^4 = (p^2)^2 \)
• \( 81 = 9^2 \)

Applying the Difference of Squares Formula

Now we can apply the formula:

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

Further Factorization

The term \( p^2 - 9 \) is also a difference of squares and can be factored further:

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

Final Factorization

Putting it all together, the complete factorization of \( p^4 - 81 \) is:

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

This gives us the fully factored form of the original expression.