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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 5
Chapter 6, Problem 5

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 5x2−6x+7(x−1)(x2+1)\(\frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)}\)

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Identify the denominator factors: here, the denominator is \( (x - 1)(x^2 + 1) \). The first factor \( (x - 1) \) is linear, and the second factor \( (x^2 + 1) \) is an irreducible quadratic.
For the linear factor \( (x - 1) \), assign a constant numerator: \( \frac{A}{x - 1} \), where \( A \) is a constant to be determined.
For the irreducible quadratic factor \( (x^2 + 1) \), assign a linear numerator: \( \frac{Bx + C}{x^2 + 1} \), where \( B \) and \( C \) are constants to be determined.
Write the partial fraction decomposition as the sum of these two fractions: \[ \frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1} \]
This is the form of the partial fraction decomposition. The next step (not required here) would be to multiply both sides by the denominator and 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 fractions into manageable parts.
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Types of Factors in Denominators

Denominators can have linear factors (like x - 1) or irreducible quadratic factors (like x² + 1). Each type requires a different form in the decomposition: linear factors correspond to constants in the numerator, while irreducible quadratics require linear expressions in the numerator.
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Formulating the Decomposition Without Solving Constants

When asked to write the form of the partial fraction decomposition without solving for constants, you set up the sum of fractions with unknown coefficients in the numerators according to the factor types. This step focuses on structure rather than finding specific values.
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Related Practice
Textbook Question

Graph each inequality. y≤(1/3)x

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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Textbook Question

In Exercises 5–18, solve each system by the substitution method. x + y = 4 y = 3x

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Textbook Question

Solve each system in Exercises 5–18.

{x+y+2z=11x+y+3z=14x+2y−z=5\(\begin{cases}\)x + y + 2z = 11 \(\x\) + y + 3z = 14 \(\x\) + 2y - z = 5\(\end{cases}\)

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Textbook Question

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (3x+16)/(x + 1) (x − 2)²

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. {y=x2−4x−10y=−x2−2x+14\(\begin{cases}\)y = x^2 - 4x - 10 \(\y\) = -x^2 - 2x + 14\(\end{cases}\)