In Exercises 65–74, factor by grouping to obtain the difference of two squares. x⁴ − x² − 2x − 1

Verified step by step guidance

1. Group the terms in pairs: \((x^4 - x^2) - (2x + 1)\).
2. Factor out the greatest common factor from each pair: \(x^2(x^2 - 1) - 1(2x + 1)\).
3. Recognize that \(x^2 - 1\) is a difference of squares: \((x^2 - 1) = (x - 1)(x + 1)\).
4. Rewrite the expression using the factored form: \(x^2(x - 1)(x + 1) - 1(2x + 1)\).
5. Check if further simplification or factoring is possible by examining the expression.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials by rearranging and grouping terms in pairs. This technique involves identifying common factors in each group, allowing for simplification. It is particularly useful when dealing with polynomials that do not have a straightforward factorization.

Difference of Two Squares

The difference of two squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is crucial for factoring expressions that can be expressed in this form, where both terms are perfect squares. Recognizing this pattern can simplify complex polynomial expressions significantly.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables raised to whole number powers and their coefficients. Understanding the structure of polynomials, including degrees and terms, is essential for performing operations like factoring. In this context, recognizing the degree of the polynomial helps in determining the appropriate factoring techniques.