Multiple Choice
Solve the equation. Then state whether it is an identity, conditional, or inconsistent equation.
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0. Review of College Algebra
Solving Linear Equations
3:22 minutesProblem 9
Textbook Question
CONCEPT PREVIEW Which of the following is the correct factorization of x⁴ - 1?A. (x² - 1) (x² + 1)B. (x² + 1) (x + 1) (x - 1)C. (x² - 1)²D. (x - 1)² (x + 1)²
Verified step by step guidance1
Recognize that the expression \(x^4 - 1\) is a difference of squares, which can be expressed as \((x^2)^2 - 1^2\).
Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x^2\) and \(b = 1\), so \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).
Notice that \(x^2 - 1\) is also a difference of squares, which can be further factored as \((x - 1)(x + 1)\).
Substitute the factorization of \(x^2 - 1\) back into the expression to get \((x - 1)(x + 1)(x^2 + 1)\).
Compare the factorized form \((x - 1)(x + 1)(x^2 + 1)\) with the given options to identify the correct answer.
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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 fundamental algebraic identity stating that a² - b² can be factored into (a - b)(a + b). This concept is crucial for factoring expressions like x⁴ - 1, as it can be recognized as a difference of squares where a = x² and b = 1.
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Factoring Quadratics
Factoring quadratics involves rewriting a quadratic expression in the form ax² + bx + c as a product of two binomials. In the context of the question, recognizing that x² - 1 can be factored further into (x - 1)(x + 1) is essential for simplifying the expression correctly.
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Polynomial Degree and Roots
The degree of a polynomial indicates the highest power of the variable, which in this case is 4 for x⁴ - 1. Understanding the roots of the polynomial, which are the values of x that make the polynomial equal to zero, helps in determining the correct factorization, as each root corresponds to a linear factor.
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