How do you factor x⁴ - y⁴?

To factor the expression \(x^4 - y^4\), we can recognize it as a difference of squares. The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). In this case, we can rewrite \(x^4 - y^4\) as follows:

Step 1: Identify Squares

Notice that \(x^4\) is \((x^2)^2\) and \(y^4\) is \((y^2)^2\). Thus, we can express the original equation as:

\( (x^2)^2 - (y^2)^2 \)

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, we can factor this into:

\( (x^2 - y^2)(x^2 + y^2) \)

Step 3: Further Factorization

The term \(x^2 - y^2\) can also be factored further since it is again a difference of squares:

\(x^2 - y^2 = (x - y)(x + y)\)

Final Factored Form

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

\( (x - y)(x + y)(x^2 + y^2) \)

This shows the expression fully factored into simpler components.