Fill in the blanks. The process of writing a rational expression as the sum or difference of two or more simpler rational expressions is called ________ ________ ________.
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Table of Contents
A.1
Real Numbers and Their Properties
A.2
Exponents and Radicals
A.3
Polynomials and Factoring
A.4
Rational Expressions
A.5
Solving Equations
A.6
Linear Inequalities in One Variable
A.7
Errors and the Algebra of Calculus
1
Functions and Their Graphs
1-3
Cumulative Test for Chapters 13
1.1
Rectangular Coordinates
1.10
Mathematical Modeling and Variation
1.2
Graphs of Equations
1.3
Linear Equations in Two Variables
1.4
Functions
1.5
Analyzing Graphs of Functions
1.6
A Library of Parent Functions
1.7
Transformations of Functions
1.8
Combinations of Functions: Composite Functions
1.9
Inverse Functions
2
Polynomial and Rational Functions
2.1
Quadratic Functions and Models
2.2
Polynomial Functions of Higher Degree
2.3
Polynomial and Synthetic Division
2.4
Complex Numbers
2.5
Zeros of Polynomial Functions
2.6
Rational Functions
2.7
Nonlinear Inequalities
3
Exponential and Logarithmic Functions
3.1
Exponential Functions and Their Graphs
3.2
Logarithmic Functions and Their Graphs
3.3
Properties of Logarithms
3.4
Exponential and Logarithmic Equations
3.5
Exponential and Logarithmic Models
4
Trigonometry
4.1
Radian and Degree Measure
4.2
Trigonometric Functions: The Unit Circle
4.3
Right Triangle Trigonometry
4.4
Trigonometric Functions of Any Angle
4.5
Graphs of Sine and Cosine Functions
4.6
Graphs of Other Trigonometric Functions
4.7
Inverse Trigonometric Functions
4.8
Applications and Models
5
Analytic Trigonometry
5.1
Using Fundamental Identities
5.2
Verifying Trigonometric Identities
5.3
Solving Trigonometric Equations
5.4
Sum and Difference Formulas
5.5
Multiple Angle and Product-to-Sum Formulas
6
Additional Topics in Trigonometry
6.1
Law of Sines
6.2
Law of Cosines
6.3
Vectors in the Plane
6.4
Vectors and Dot Products
6.5
Trigonometric Form of a Complex Number
7
Systems of Equations and Inequalities
7.1
Linear and Nonlinear Systems of Equations
7.2
Two-Variable Linear Systems
7.3
Multivariable Linear Systems
7.4
Partial Fractions
7.5
Systems of Inequalities
7.6
Linear Programming
8
Matrices and Determinants
8.1
Matrices and Systems of Equations
8.2
Operations with Matrices
8.3
The Inverse of a Square Matrix
8.4
The Determinant of a Square Matrix
8.5
Applications of Matrices and Determinants
9
Sequences, Series, and Probability
9.1
Sequences and Series
9.2
Arithmetic Sequences and Partial Sums
9.3
Geometric Sequences and Series
9.4
Mathematical Induction
9.5
The Binomial Theorem
9.6
Counting Principles
9.7
Probability
10
Topics in Analytic Geometry
10.1
Lines
10.2
Introduction to Conics: Parabolas
10.3
Ellipses
10.4
Hyperbolas
10.5
Rotation of Conics
10.6
Parametric Equations
10.7
Polar Coordinates
10.8
Graphs of Polar Equations
10.9
Polar Equations of Conics
Textbook Solutions for Precalculus
Chapter 7.4 Problem 53
Question
In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
Solution
The first step in solving 7.4 problem number 57 trying to solve the problem we have to refer to the textbook question: In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
From the textbook chapter Partial Fractions you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Precalculus 7
Author
Ron Larson, Robert P. Hostetler
ISBN
9780618643448
In Exercises 5356, (a) write the partial fraction decomposition of the rational
Chapter 7.4 textbook questions
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Chapter 7: Problem 1 Precalculus 7
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Chapter 7: Problem 2 Precalculus 7
Fill in the blanks. If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ________.
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Chapter 7: Problem 3 Precalculus 7
Fill in the blanks. In order to find the partial fraction decomposition of a rational expression, the denominator must be completely factored into ________ factors of the form and ________ factors of the form which are ________ over the rationals.
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Chapter 7: Problem 4 Precalculus 7
Fill in the blanks. The ________ ________ is derived after multiplying each side of the partial fraction decomposition form by the least common denominator.
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Chapter 7: Problem 5 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 6 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 7 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 8 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 9 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 10 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 11 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 12 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 13 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 14 Precalculus 7
In Exercises 514, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
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Chapter 7: Problem 15 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 16 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 17 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 18 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 19 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 20 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 21 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 22 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 23 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 24 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 25 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 26 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 27 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 28 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 29 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 30 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 31 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 32 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 33 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 34 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 35 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 36 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 37 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 38 Precalculus 7
In Exercises 1538, write the partial fraction decomposition of the rational expression. Check your result algebraically.
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Chapter 7: Problem 39 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 40 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 41 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 42 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 43 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 44 Precalculus 7
In Exercises 3944, write the partial fraction decomposition of the improper rational expression.
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Chapter 7: Problem 45 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 46 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 47 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 48 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 49 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 50 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 51 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 52 Precalculus 7
In Exercises 4552, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically.
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Chapter 7: Problem 53 Precalculus 7
In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
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Chapter 7: Problem 54 Precalculus 7
In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
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Chapter 7: Problem 55 Precalculus 7
In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
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Chapter 7: Problem 56 Precalculus 7
In Exercises 5356, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
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Chapter 7: Problem 57 Precalculus 7
Thermodynamics The magnitude of the range of exhaust temperatures (in degrees Fahrenheit) in an experimental diesel engine is approximated by the model R _ 0 < x 1 where is the relative load (in foot-pounds). (a) Write the partial fraction decomposition of the equation. (b) The decomposition in part (a) is the difference of two fractions. The absolute values of the terms give the expected maximum and minimum temperatures of the exhaust gases for different loads. Write the equations for Ymax and Ymin. (c) Use a graphing utility to graph each equation from part (b) in the same viewing window. (d) Determine the expected maximum and minimum temperatures for a relative load of 0.5.
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Chapter 7: Problem 58 Precalculus 7
Writing Describe two ways of solving for the constants in a partial fraction decomposition. True or False?
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Chapter 7: Problem 59 Precalculus 7
True or False? In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. For the rational expression the partial fraction decomposition is of the form
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Chapter 7: Problem 60 Precalculus 7
True or False? In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. When writing the partial fraction decomposition of the expression the first step is to factor the denominator. a
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Chapter 7: Problem 61 Precalculus 7
In Exercises 6164, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically.
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Chapter 7: Problem 62 Precalculus 7
In Exercises 6164, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically.
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Chapter 7: Problem 63 Precalculus 7
In Exercises 6164, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically.
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Chapter 7: Problem 64 Precalculus 7
In Exercises 6164, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically.
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Chapter 7: Problem 70 Precalculus 7
In Exercises 6570, sketch the graph of the function. f _x_ _ 3x _ 1 x2 _ 4x _ 12
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