Explanation Given information: The function is 3 x 3 + 22 x 2 + 53 x + 41 ( x + 2 ) 2 ( x + 3 ) 2 . Concept used: Consider p ( x ) be any ration function that can be expressed as, p ( x ) = P ( x ) Q ( x ) Also consider that Q ( x ) has linear factor ( m x + n ) that is repeated k times such that ( m x + n ) k is a factor of the denominator Q ( x ) . Then partial fraction decomposition of the rational function p ( x ) can be expressed as, P ( x ) Q ( x ) = A 1 m x + n + A 2 ( m x + n ) 2 + A 3 ( m x + n ) 3 + ⋯ + A k ( m x + n ) k Here, A 1 , A 2 , A 3 , … , A k are constants. Calculation: The given function is 3 x 3 + 22 x 2 + 53 x + 41 ( x + 2 ) 2 ( x + 3 ) 2 . Compare the above function with the form P ( x ) Q ( x ) to obtain the value of P ( x ) and Q ( x ) as, P ( x ) = 3 x 3 + 22 x 2 + 53 x + 41 Q ( x ) = ( x + 2 ) 2 ( x + 3 ) 2 The denominator Q ( x ) of the function is ( x + 2 ) 2 ( x + 3 ) 2 . The factors of Q ( x ) are ( x + 2 ) , ( x + 2 ) 2 , ( x + 3 ) and ( x + 3 ) 2 . Therefore, the partial fraction decomposition of the function is, 3 x 3 + 22 x 2 + 53 x + 41 ( x + 2 ) 2 ( x + 3 ) 2 = A x + 2 + B ( x + 2 ) 2 + C ( x + 3 ) + D ( x + 3 ) 2 …… (1) Here, A , B , C and D are constants. The common denominator of ( x + 2 ) , ( x + 2 ) 2 , ( x + 3 ) and ( x + 3 ) 2 is ( x + 2 ) 2 ( x + 3 ) 2 . Now, multiply Right Hand Side of equation (1) by ( x + 2 ) 2 ( x + 3 ) 2 as, 3 x 3 + 22 x 2 + 53 x + 41 = { A ( x + 2 ) ( x + 3 ) 2 + B ( x + 3 ) 2 + C ( x + 2 ) 2 ( x + 3 ) + D ( x + 2 ) 2 } = { A ( x + 2 ) ( x 2 + 6 x + 9 ) + B ( x 2 + 6 x + 9 ) + C ( x 2 + 4 x + 4 ) ( x + 3 ) + D ( x 2 + 4 x + 4 ) } = { A ( x 3 + 2 x 2 + 6 x 2 + 12 x + 9 x + 18 ) + B ( x 2 + 6 x + 9 ) + C ( x 3 + 3

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Quadratic Equation

When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations …

Demand and Supply Function

The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to …

Videos

Question

Chapter 5.3, Problem 35E

To determine

To find:

The partial fraction decomposition of the function 3x3+22x2+53x+41( x+2)2( x+3)2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.3, Problem 34E

Chapter 5.3, Problem 36E

Students have asked these similar questions

This is discrete mathematics

Write the partial fraction decomposition of the improper rational expression. x3 - 5x2 - 7x – 10 x² - 6x - 7

Chapter 5 Solutions

College Algebra

Ch. 5.1 - The system of equations {2x+3y=75xy=9} is a system...Ch. 5.1 - A system of equation in two variable can be solved...Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...

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 partial fraction decomposition of the function 3 x 3 + 22 x 2 + 53 x + 41 ( x + 2 ) 2 ( x + 3 ) 2 .

Solution Summary: The author explains the partial tion decomposition of the rational function p(x).

Concept explainers

Videos

To find:

The partial fraction decomposition of the function 3x3+22x2+53x+41( x+2)2( x+3)2.

Want to see the full answer?

Chapter 5 Solutions

The partial fraction decomposition of the function 3 x 3 + 22 x 2 + 53 x + 41 ( x + 2 ) 2 ( x + 3 ) 2 .

Solution Summary: The author explains the partial tion decomposition of the rational function p(x).

Concept explainers

Videos

To find: The partial fraction decomposition of the function 3x3+22x2+53x+41( x+2)2( x+3)2.

Want to see the full answer?

Chapter 5 Solutions

To find:

The partial fraction decomposition of the function 3x3+22x2+53x+41( x+2)2( x+3)2.