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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 10
Capitolo 6, Problema 10

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

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Identify the form of the rational expression. Here, the denominator is a product of two linear factors: x(x-1).
Set up the partial fraction decomposition by expressing the rational expression as a sum of fractions with unknown constants in the numerators over each linear factor: \(\frac{1}{x(x-1)}\) = \(\frac{A}{x}\) + \(\frac{B}{x-1}\), where A and B are constants to be determined.
Multiply both sides of the equation by the common denominator x(x-1) to clear the denominators: 1 = A(x-1) + Bx.
Expand the right side: 1 = A x - A + B x, then combine like terms: 1 = (A + B) x - A.
Equate the coefficients of corresponding powers of x on both sides to form a system of equations: For the x term, 0 = A + B; for the constant term, 1 = -A. These equations can be solved to find A and B.

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Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.
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Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.
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Factoring Polynomials

Factoring polynomials means rewriting a polynomial as a product of its factors. Recognizing and factoring denominators into linear or irreducible quadratic factors is crucial for setting up the correct form of partial fractions.
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