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 R.3.9
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)² D. (x - 1)² (x + 1)²
Verified step by step guidance1
Recognize that the expression \(x^4 - 1\) is a difference of squares, since it can be written as \((x^2)^2 - 1^2\).
Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x^2\) and \(b = 1\). This gives \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).
Notice that \(x^2 - 1\) is itself a difference of squares and can be further factored as \((x - 1)(x + 1)\).
Combine the factors to express the full factorization as \((x - 1)(x + 1)(x^2 + 1)\).
Compare this factorization with the given options to identify which one matches the correct factorization.
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Video duration:
3mKey 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 a² - b² can be factored into (a - b)(a + b). For example, x⁴ - 1 can be seen as (x²)² - 1², allowing it to be factored into (x² - 1)(x² + 1).
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Further Factorization of Quadratic Expressions
Some quadratic expressions, like x² - 1, can be further factored if they are also differences of squares. Since x² - 1 = (x - 1)(x + 1), recognizing this allows complete factorization of higher-degree polynomials into linear factors.
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Understanding Polynomial Factorization and Exponents
Polynomial factorization involves breaking down expressions into products of simpler polynomials. Recognizing powers and exponents, such as x⁴ being (x²)², helps in applying factoring formulas correctly and identifying equivalent factorizations.
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