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 25
Capitolo 6, Problema 25

Write the partial fraction decomposition of each rational expression. x2+2x+7/x(x − 1)2

Guida verificata passo dopo passo
1
Identify the form of the denominator and write the general form of the partial fraction decomposition. Since the denominator is \(x(x - 1)^2\), the decomposition will include terms for the linear factor \(x\) and the repeated linear factor \((x - 1)^2\).
Set up the partial fraction decomposition as: \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides of the equation by the common denominator \(x(x - 1)^2\) to clear the fractions, resulting in: $x^2 + 2x + 7 = A(x - 1)^2 + Bx(x - 1) + Cx$.
Expand the right-hand side by applying the distributive property and simplifying each term: expand \((x - 1)^2\), multiply terms, and combine like terms.
Equate the coefficients of corresponding powers of \(x\) on both sides of the equation to form a system of linear equations in \(A\), \(B\), and \(C\). 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:
8m

Concetti chiave

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

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.
Video consigliato:
4:07
Decomposition of Functions

Factorization of Denominators

Understanding how to factor the denominator is essential for setting up partial fractions. In this problem, the denominator x(x − 1)² includes a linear factor and a repeated linear factor, which affects the form of the decomposition by requiring terms for each power of the repeated factor.
Video consigliato:
02:58
Rationalizing Denominators

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of partial fractions, you equate the numerators and solve for unknown coefficients. This involves multiplying both sides by the common denominator and comparing coefficients of corresponding powers of x to form a system of equations.
Video consigliato:
5:02
Solving Logarithmic Equations
Pratica correlata
Domanda del libro di testo

Solve each system in Exercises 25–26. {x+26−y+43+z2=0x+12+y−12−z4=92x−54+y+13+z−22=194\(\begin{cases}\) \(\frac{x + 2}{6}\) - \(\frac{y + 4}{3}\) + \(\frac{z}{2}\) = 0 \\ \(\frac{x + 1}{2}\) + \(\frac{y - 1}{2}\) - \(\frac{z}{4}\) = \(\frac{9}{2}\) \\ \(\frac{x - 5}{4}\) + \(\frac{y + 1}{3}\) + \(\frac{z - 2}{2}\) = \(\frac{19}{4}\) \(\end{cases}\)

603
views
Domanda del libro di testo

Graph each inequality. y≥log2(x+1)

585
views
Domanda del libro di testo

In Exercises 19–30, solve each system by the addition method. 4x + 3y = 15 2x - 5y = 1

720
views
Domanda del libro di testo

In Exercises 25–35, solve each system by the method of your choice. This is a piecewise function, refer to textbook problem.

536
views
Domanda del libro di testo

In Exercises 19–30, solve each system by the addition method. 3x - 4y = 11 2x + 3y = - 4

759
views
Domanda del libro di testo

Solve each system in Exercises 25–26. {x+32−y−12+z+24=32x−52+y+13−z4=−256x−34−y+12+z−32=−52\(\begin{cases}\) \(\frac{x + 3}{2}\) - \(\frac{y - 1}{2}\) + \(\frac{z + 2}{4}\) = \(\frac{3}{2}\) \\ \(\frac{x - 5}{2}\) + \(\frac{y + 1}{3}\) - \(\frac{z}{4}\) = - \(\frac{25}{6}\) \\ \(\frac{x - 3}{4}\) - \(\frac{y + 1}{2}\) + \(\frac{z - 3}{2}\) = - \(\frac{5}{2}\) \(\end{cases}\)

630
views