Table of contents
- 0. Review of Algebra4h 16m
- 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 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Factoring Polynomials
Problem 78dLial - 13th Edition
Textbook Question
Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]
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1
Identify the complex fraction: .
Recall that dividing by a fraction is the same as multiplying by its reciprocal. Rewrite the expression as: .
Multiply the numerators and the denominators: .
Factor the numerator using the difference of squares: .
Factor the denominator using the difference of cubes: .
Recommended similar problem, with video answer:
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Fractions
A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the inner fractions and then simplifies the overall expression. Understanding how to manipulate these nested fractions is crucial for effective simplification.
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Complex Conjugates
Difference of Squares
The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is essential when simplifying expressions like x² - y², as it allows us to factor the expression into linear factors, making it easier to work with in complex fractions.
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Solving Quadratic Equations by Completing the Square
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of its factors. This is a fundamental skill in algebra, as it simplifies expressions and helps in solving equations. In the context of the given problem, recognizing how to factor x³ - y³ and x² - y² will facilitate the simplification of the complex fraction.
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Introduction to Factoring Polynomials
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Related Practice
Textbook Question
In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers.
x² - 4
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