Factor each polynomial. See Examples 5 and 6. x4-16

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 61

Chapter 1, Problem 61

Factor each polynomial. See Examples 5 and 6. x⁴-16

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Recognize that the polynomial $x^4 - 16$ is a difference of squares because it can be written as $(x^2)^2 - 4^2$.

Apply the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$, where $a = x^2$ and $b = 4$. This gives us $(x^2 - 4)(x^2 + 4)$.

Notice that $x^2 - 4$ is itself a difference of squares and can be factored further as $(x - 2)(x + 2)$.

The term $x^2 + 4$ is a sum of squares, which does not factor over the real numbers, so it remains as is.

Combine all factors to express the fully factored form: $(x - 2)(x + 2)(x^2 + 4)$.

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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 used when an expression is in the form a^2 - b^2. It factors into (a - b)(a + b). Recognizing this pattern helps simplify polynomials like x^4 - 16 by expressing terms as squares.

Factoring Higher Powers

Polynomials with higher powers, such as x^4, can often be rewritten as powers of squares (e.g., (x^2)^2). This allows the use of difference of squares repeatedly to factor completely, breaking down complex expressions step-by-step.

Polynomial Factoring Strategy

Factoring polynomials involves identifying patterns and applying appropriate methods like difference of squares, grouping, or special formulas. A systematic approach ensures complete factorization, which is essential for solving or simplifying polynomial expressions.

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