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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 7
Chapter 6, Problem 7

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x3+x2(x2+4)2\(\frac{x^3 + x^2}{(x^2 + 4)^2}\)

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1
Identify the rational expression given: \(\frac{x^3 + x^2}{(x^2 + 4)^2}\).
Recognize that the denominator is a repeated irreducible quadratic factor: \((x^2 + 4)^2\).
For a repeated irreducible quadratic factor \((ax^2 + bx + c)^n\), the partial fraction decomposition includes terms of the form \(\frac{Bx + C}{ax^2 + bx + c}\), \(\frac{Dx + E}{(ax^2 + bx + c)^2}\), and so on, up to the power \(n\).
Write the general form of the partial fraction decomposition for this problem as: \(\frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2}\), where \(B\), \(C\), \(D\), and \(E\) are constants to be determined.
Note that since the numerator degree is less than the denominator degree, no polynomial division is necessary before setting up the partial fractions.

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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, making integration or other operations easier. It involves breaking down a complex rational expression into a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.
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Repeated Quadratic Factors

When the denominator contains a repeated irreducible quadratic factor, such as (x² + 4)², the decomposition includes terms with the quadratic factor raised to each power up to its multiplicity. For example, terms with denominators (x² + 4) and (x² + 4)² appear, each with linear numerators.
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Form of Numerators in Partial Fractions

For linear factors, numerators are constants; for irreducible quadratic factors, numerators are linear expressions (ax + b). This ensures the decomposition can represent any polynomial numerator of lower degree than the denominator factor, allowing for a complete and accurate breakdown.
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Related Practice
Textbook Question

In Exercises 5–18, solve each system by the substitution method. {x+3y=8y=2x−9\(\begin{cases}\)x + 3y = 8 \(\y\) = 2x - 9\(\end{cases}\)

Textbook Question

Graph each inequality. y>2x−1

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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 1–18, solve each system by the substitution method.{x2+y2=25x−y=1\(\begin{cases}\)x^2 + y^2 = 25 \(\x\) - y = 1\(\end{cases}\)

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Textbook Question

Solve each system in Exercises 5–18. {4x−y+2z=11x+2y−z=−12x+2y−3z=−1\(\begin{cases}\)4x - y + 2z = 11 \(\x\) + 2y - z = -1 \\2x + 2y - 3z = -1\(\end{cases}\)

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.

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