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

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Chapter 5.3, Problem 35E

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

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Chapter 5.3, Problem 34E

Chapter 5.3, Problem 36E

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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).

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

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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).

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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.