Skip to main content
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 8
Chapter 6, Problem 8

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. (7x2 -9x+3)/(x2+7)2

Verified step by step guidance
1
Identify the denominator and factor it if possible. Here, the denominator is x² + 72, which is already factored as (x^2 + 7)^2.
Since the denominator is a repeated irreducible quadratic factor, the partial fraction decomposition will include terms for each power of this factor up to 2.
For each power of the irreducible quadratic factor (x^2 + 7), write a numerator that is a linear polynomial: for the first power, use Ax + B, and for the second power, use Cx + D.
Set up the partial fraction decomposition as the sum of these terms: \(\frac{Ax + B}{x^2 + 7}\) + \(\frac{Cx + D}{(x^2 + 7)^2}\).
This form represents the partial fraction decomposition of the given rational expression without solving for the constants A, B, C, and D.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
4:07
Decomposition of Functions

Repeated Quadratic Factors

When the denominator contains a repeated irreducible quadratic factor, such as (x² + 7)², the decomposition includes terms with the quadratic factor raised to increasing powers. Each term has a linear numerator for quadratic factors, reflecting the factor's multiplicity.
Recommended video:
06:08
Solving Quadratic Equations by Factoring

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 the original rational function accurately before solving for constants.
Recommended video:
Guided course
05:45
Radical Expressions with Fractions
Related Practice
Textbook Question

Graph each inequality. y>2x−1

2
views
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.

3
views
Textbook Question

A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?

1
views
Textbook Question

In Exercises 5–18, solve each system by the substitution method. {x=4y−2x=6y+8\(\begin{cases}\)x = 4y - 2 \(\x\) = 6y + 8\(\end{cases}\)

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

Write the partial fraction decomposition of each rational expression. x/(x-2)(x-3)