Explanation Calculation: Consider the expression, x 2 + 5 x + 1 x 2 − 2 x + 3 . Note that, the highest power of the variable is the degree of an expression. Thus, the degree of the numerator is 2 and the degree of the denominator is 3 . The given rational expression is a proper fraction because the degree of the numerator is less than the degree of the denominator. The denominator is in the factorised from as x + 1 x 2 − 2 x + 3 . Consider, degree of the factor x + 1 is 1 , which is linear, and that of the factor x 2 − 2 x + 3 is 2 , which is quadratic. Since the factor x + 1 is linear and x 2 − 2 x + 3 is quadratic, write one partial fraction with its numerator as an unknown constant for the linear factor and another partial fraction with an unknown linear numerator for the quadratic factor. Thus, x 2 + 5 x + 1 x 2 − 2 x + 3 = A x + 1 + B x + C x 2 − 2 x + 3 The least common denominator (LCD) of the terms on the right side is x + 1 x 2 − 2 x + 3 . So, in order to make the denominators same, A x + 1 + B x + C x 2 − 2 x + 3 = A x 2 − 2 x + 3 x + 1 x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 = A x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 Thus, x 2 + 5 x + 1 x 2 − 2 x + 3 = A x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 . Now, the denominators on both sides are same. So, multiply both sides by the LCD to get the basic equation. x 2 + 5 = A x 2 − 2 x + 3 + B x + C x + 1 To solve the basic equation for the unknowns A , B and C , simplify the basic equation. x 2 + 5 = A x 2 − 2 x + 3 + B x + C x + 1 = A x 2 − 2 A x + 3 A + B x 2 + B x + C x + C = A + B x 2 + − 2 A + B + C x + 3 A + C Equate the like terms on either sides of the last equation to get. 1 = A + B …… (1) 0 = − 2 A + B + C …… (2) 5 = 3 A + C …… (3) Multiply equation (1) by − 1 to get − 1 = − A − B …… (4) Add equations (2), (3) and (4) to get

10th Edition

ISBN: 9781337282291

Author: Ron Larson

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

Videos

Question

Chapter 6.4, Problem 35E

To determine

To calculate: The form of partial fraction decomposition of the rational expression x2+5x+1x2−2x+3 and check the answer algebraically.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.4, Problem 34E

Chapter 6.4, Problem 36E

Students have asked these similar questions

Assume {u1, U2, us} spans R³. Select the best statement. A. {U1, U2, us, u4} spans R³ unless u is the zero vector. B. {U1, U2, us, u4} always spans R³. C. {U1, U2, us, u4} spans R³ unless u is a scalar multiple of another vector in the set. D. We do not have sufficient information to determine if {u₁, u2, 43, 114} spans R³. OE. {U1, U2, 3, 4} never spans R³. F. none of the above

Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the above

Let H = span {u, v}. For each of the following sets of vectors determine whether H is a line or a plane. Select an Answer u = 3 1. -10 8-8 -2 ,v= 5 Select an Answer -2 u = 3 4 2. + 9 ,v= 6

Chapter 6 Solutions

College Algebra

Ch. 6.1 - Solve the system of equations. xy=05x3y=6 Ch. 6.1 - A total of $25,000 is invested in two funds paying...Ch. 6.1 - Solve the system of equations. 2x+y=5x2y+3x=1 Ch. 6.1 - Solve the system of equations. 2x+y=32x2+4xy2=0 Ch. 6.1 - Prob. 5ECP Ch. 6.1 - Prob. 6ECP Ch. 6.1 - Prob. 7ECP Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Fill in the blanks. Graphically, solutions of a...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The form of partial fraction decomposition of the rational expression x 2 + 5 x + 1 x 2 − 2 x + 3 and check the answer algebraically.

Solution Summary: The author calculates the form of partial tion decomposition of the rational expression, x2+5(x+1) and check the answer algebraically.

Concept explainers

Videos

To calculate: The form of partial fraction decomposition of the rational expression x2+5x+1x2−2x+3 and check the answer algebraically.

Want to see the full answer?

Chapter 6 Solutions