Textbook Question
In Exercises 29–42, solve each system by the method of your choice.
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 6, Problem 31
Write the partial fraction decomposition of each rational expression. 5x2+6x+3/(x + 1)(x² + 2x + 2)
Verified step by step guidance1
Identify the form of the denominator. Here, the denominator is \( (x + 1)(x^2 + 2x + 2) \), which consists of a linear factor \( (x + 1) \) and an irreducible quadratic factor \( (x^2 + 2x + 2) \).
Set up the partial fraction decomposition with unknown constants. For the linear factor \( (x + 1) \), use a constant numerator \( A \). For the irreducible quadratic factor \( (x^2 + 2x + 2) \), use a linear numerator \( Bx + C \). So, write:
\[ \frac{5x^2 + 6x + 3}{(x + 1)(x^2 + 2x + 2)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 2x + 2} \]
Multiply both sides of the equation by the denominator \( (x + 1)(x^2 + 2x + 2) \) to clear the fractions:
\[ 5x^2 + 6x + 3 = A(x^2 + 2x + 2) + (Bx + C)(x + 1) \]
Expand the right-hand side and then collect like terms (powers of \( x \)) to form an equation where the coefficients of corresponding powers of \( x \) on both sides are equal. This will give a system of equations to solve for \( A \), \( B \), and \( C \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 expressions into manageable parts.
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Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its factors. Recognizing linear factors like (x + 1) and irreducible quadratic factors like (x² + 2x + 2) is essential for setting up the correct form of partial fractions.
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Setting Up Partial Fractions for Linear and Quadratic Factors
When decomposing, linear factors correspond to terms with constants in the numerator (A/(x+1)), while irreducible quadratic factors require linear expressions in the numerator (Bx + C)/(x² + 2x + 2). Correctly assigning these forms is crucial for solving the decomposition.
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Related Practice
Textbook Question
In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x = 9-2y x + 2y = 13
Textbook Question
Graph the solution set of each system of inequalities or indicate that the system has no solution.
Textbook Question
In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. y = 3x - 5 21x - 35 = 7y
Textbook Question
In Exercises 29–42, solve each system by the method of your choice.
Textbook Question
Graph the solution set of each system of inequalities or indicate that the system has no solution.