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 82cLial - 13th Edition
Textbook Question
Simplify each complex fraction. [ 6/(x^2-25) + x ] / [ 1/(x - 5) ]
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1
Identify the complex fraction: .
Recognize that is a difference of squares, which can be factored as .
Rewrite the complex fraction: .
To simplify, multiply the numerator and the denominator by to eliminate the fractions.
Simplify the expression by distributing and combining like terms, then simplify the resulting expression.
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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 rewrites the complex fraction as a single fraction. This process often involves algebraic manipulation to eliminate the nested fractions.
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Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the given question, recognizing that x^2 - 25 is a difference of squares allows us to factor it as (x - 5)(x + 5), which simplifies the overall expression and aids in further simplification.
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Common Denominators
A common denominator is a shared multiple of the denominators of two or more fractions, allowing for the addition or subtraction of those fractions. In the context of the given problem, finding a common denominator is essential for combining the fractions in the numerator before dividing by the fraction in the denominator, ultimately leading to a simplified result.
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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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