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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2:19 minutes

Problem 93

Textbook Question

In Exercises 75–94, factor using the formula for the sum or difference of two cubes. (x − y)³ − y³

Verified step by step guidance

Identify the expression as a difference of cubes: \((x - y)^3 - y^3\).

Recognize the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Assign \(a = x - y\) and \(b = y\) to match the formula.

Apply the formula: \((x - y) - y)((x - y)^2 + (x - y)y + y^2)\).

Simplify the expression: \((x - 2y)((x - y)^2 + (x - y)y + y^2)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Difference of Cubes

The difference of cubes is a specific algebraic identity that states a³ - b³ = (a - b)(a² + ab + b²). This formula allows us to factor expressions where one term is the cube of a variable and the other is the cube of another variable, facilitating simplification and solving of equations.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Understanding various factoring techniques, including the difference of cubes, is essential for simplifying polynomials and solving algebraic equations effectively.

Cubic Expressions

Cubic expressions are polynomial expressions of degree three, typically in the form ax³ + bx² + cx + d. Recognizing the structure of cubic expressions is crucial for applying appropriate factoring methods, such as the difference of cubes, to simplify or solve them.