Skip to main content
Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 17
Capitolo 6, Problema 17

Write the partial fraction decomposition of each rational expression. 4x2+13x-9/x (x − 1)(x+3)

Guida verificata passo dopo passo
1
Identify the form of the rational expression. Since the denominator is factored as \(x(x - 1)(x + 3)\), and all factors are linear and distinct, the partial fraction decomposition will have terms of the form \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
Set up the equation by expressing the given rational expression as a sum of partial fractions: \(\frac{4x^{2} + 13x - 9}{x(x - 1)(x + 3)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
Multiply both sides of the equation by the common denominator \(x(x - 1)(x + 3)\) to clear the denominators: \(4x^{2} + 13x - 9 = A(x - 1)(x + 3) + B x (x + 3) + C x (x - 1)\).
Expand the right-hand side by multiplying out each term: - \(A(x - 1)(x + 3) = A(x^{2} + 2x - 3)\) - \(B x (x + 3) = B(x^{2} + 3x)\) - \(C x (x - 1) = C(x^{2} - x)\) Then combine all terms to get a polynomial in standard form.
Group like terms (powers of \(x\)) on the right side and equate the coefficients of corresponding powers of \(x\) from both sides to form a system of equations. Solve this system to find the values of \(A\), \(B\), and \(C\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
13m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including factoring and simplifying, is essential before performing partial fraction decomposition.
Video consigliato:
02:58
Rationalizing Denominators

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.
Video consigliato:
4:07
Decomposition of Functions

Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors. For partial fraction decomposition, the denominator must be factored into linear or irreducible quadratic factors to set up the correct form of the decomposition.
Video consigliato:
07:30
Introduction to Factoring Polynomials
Pratica correlata
Domanda del libro di testo

Solve each system in Exercises 5–18. {3(2x+y)+5z=−12(x−3y+4z)=−94(1+x)=−3(z−3y)\(\begin{cases}\) 3(2x + y) + 5z = -1 \\ 2(x - 3y + 4z) = -9 \\ 4(1 + x) = -3(z - 3y) \(\end{cases}\)

652
views
Domanda del libro di testo

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

705
views
Domanda del libro di testo

Graph each inequality. (x−2)2+(y+1)2<9

634
views
Domanda del libro di testo

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

802
views
Domanda del libro di testo

In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2

725
views
Domanda del libro di testo

In Exercises 1–18, solve each system by the substitution method. {x+y=1(x−1)2+(y+2)2=10\(\begin{cases}\) x + y = 1 \\ (x - 1)^2 + (y + 2)^2 = 10 \(\end{cases}\)

564
views