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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 9
Chapter 1, Problem 9

Work each problem. Which of the following is the correct complete factorization of x4−1x^4-1?
A. (x2−1)(x2+1)(x^2-1)(x^2+1)
B. (x2+1)(x+1)(x−1)(x^2+1)(x+1)(x-1)
C. (x2−1)2(x^2-1)^2
D. (x−1)2(x+1)2(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 \((x^2)^2 - 1^2\).
Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), so \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).
Notice that \(x^2 - 1\) is itself a difference of squares and can be factored further as \((x - 1)(x + 1)\).
Combine the factors to write the complete factorization as \((x - 1)(x + 1)(x^2 + 1)\).
Compare this factorization with the given options to identify the correct one.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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).
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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.
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