How do you factor completely a⁴ - b⁴ ?

To factor the expression \(4 - b^4\), we can recognize it as a difference of squares. The expression can be rewritten as:

Step 1: Identify the Difference of Squares

We can express \(4\) as \(2^2\) and \(b^4\) as \((b^2)^2\). Thus, we have:

4 - b^4 = 2^2 - (b^2)^2

Step 2: Apply the Difference of Squares Formula

The difference of squares can be factored using the formula:

a^2 - b^2 = (a - b)(a + b)

Applying this to our expression:

(2 - b^2)(2 + b^2)

Step 3: Factor Further if Possible

Now, we can check if \(2 - b^2\) can be factored further. Notice that \(2 - b^2\) is also a difference of squares:

2 - b^2 = \sqrt{2}^2 - (b)^2

Using the difference of squares formula again:

(\sqrt{2} - b)(\sqrt{2} + b)

Final Factored Form

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

• (\sqrt{2} - b)(\sqrt{2} + b)(2 + b^2)

This is the fully factored form of the expression \(4 - b^4\).