Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

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2:31 minutes

Problem 45

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁸ − y⁸

Verified step by step guidance

Recognize that the expression \(x^8 - y^8\) is a difference of two powers.

Recall the formula for factoring a difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).

Notice that \(x^8 - y^8\) can be rewritten as \((x^4)^2 - (y^4)^2\), which is a difference of squares.

Apply the difference of squares formula: \((x^4 - y^4)(x^4 + y^4)\).

Factor \(x^4 - y^4\) further using the difference of squares: \((x^2 - y^2)(x^2 + y^2)\).

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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 polynomials that can be expressed as the difference between two perfect squares, which is applicable in the given polynomial x⁸ - y⁸.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. Understanding how to identify common factors, apply special identities, and break down higher-degree polynomials is essential for completely factoring expressions like x⁸ - y⁸.

Prime Polynomials

A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing when a polynomial is prime is important in algebra, as it determines whether further factorization is possible, which is relevant when analyzing the polynomial x⁸ - y⁸.