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

Verified step by step guidance

1. Recognize that the polynomial \(x^4 - 16\) is a difference of squares because it can be written as \((x^2)^2 - 4^2\).
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 \((x^2 - 4)(x^2 + 4)\).
3. Notice that \(x^2 - 4\) is itself a difference of squares and can be factored further as \((x - 2)(x + 2)\).
4. The term \(x^2 + 4\) is a sum of squares, which does not factor over the real numbers, so it remains as is.
5. Combine all factors to express the fully factored form: \((x - 2)(x + 2)(x^2 + 4)\).

Key Concepts

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.