Work each problem. Which of the following is the correct complete factorization of $x^4-1$?
A. $(x^2-1)(x^2+1)$
B. $(x^2+1)(x+1)(x-1)$
C. $(x^2-1{)}^2$
D. $(x-1{)}^2(x+1{)}^2$

Verified step by step guidance

1. Recognize that the expression \(x^4 - 1\) is a difference of squares because it can be written as \((x^2)^2 - 1^2\).
2. Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), so \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).
3. Notice that \(x^2 - 1\) is itself a difference of squares and can be factored further as \((x - 1)(x + 1)\).
4. Combine the factors to write the complete factorization as \((x - 1)(x + 1)(x^2 + 1)\).
5. Compare this factorization with the given options to identify the correct one.

Key Concepts

Difference of Squares

The difference of squares is a factoring technique where an expression of the form a^2 - b^2 can be factored into (a - b)(a + b). This is essential for factoring x^4 - 1 because it can be seen as (x^2)^2 - 1^2.

Factoring Quadratic Expressions

After applying the difference of squares once, the resulting quadratic expressions may be factored further if possible. For example, x^2 - 1 can be factored again using the difference of squares into (x - 1)(x + 1).

Complete Factorization

Complete factorization means breaking down a polynomial into irreducible factors over the real numbers. For x^4 - 1, this involves factoring repeatedly until no further factorization is possible, ensuring the expression is fully simplified.