Explanation Given information: The function is 3 x 2 + 5 x − 13 ( 3 x + 2 ) ( x 2 − 4 x + 4 ) . 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 2 + 5 x − 13 ( 3 x + 2 ) ( x 2 − 4 x + 4 ) . 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 2 + 5 x − 13 Q ( x ) = ( 3 x + 2 ) ( x 2 − 4 x + 4 ) The denominator Q ( x ) of the function is ( 3 x + 2 ) ( x 2 − 4 x + 4 ) . The factorization of denominator Q ( x ) can be calculated as, Q ( x ) = ( 3 x + 2 ) ( x 2 − 4 x + 4 ) = ( 3 x + 2 ) ( x 2 − 2 ⋅ 2 ⋅ x + 2 2 ) = ( 3 x + 2 ) ( x − 2 ) 2 Therefore, the factors of Q ( x ) are ( 3 x + 2 ) , ( x − 2 ) and ( x − 2 ) 2 . Therefore, the partial fraction decomposition of the function is, 3 x 2 + 5 x − 13 ( 3 x + 2 ) ( x 2 − 4 x + 4 ) = A 3 x + 2 + B x − 2 + C ( x − 2 ) 2 …… (1) Here, A , B and C are constants. The common denominator of ( 3 x + 2 ) , ( x − 2 ) and ( x − 2 ) 2 is ( 3 x + 2 ) ( x − 2 ) 2 . Now, multiply Right Hand Side of equation (1) by ( 3 x + 2 ) ( x − 2 ) 2 as, 3 x 2 + 5 x − 13 = A ( x − 2 ) 2 + B ( x − 2 ) ( 3 x + 2 ) + C ( 3 x + 2 ) = A ( x 2 − 4 x + 4 ) + B ( 3 x 2 − 6 x + 2 x − 4 ) + C ( 3 x + 2 ) = A x 2 − 4 A x + 4 A + 3 B x 2 − 4 B x − 4 B + 3 C x + 2 C = ( A + 3 B ) x 2 + ( − 4 A − 4 B + 3 C ) x + 4 A − 4 B + 2 C Two polynomials are equal only if the coefficients of two polynomials are equal

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 5.3, Problem 28E

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The partial fraction decomposition of the function 3x2+5x−13(3x+2)(x2−4x+4).

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Chapter 5.3, Problem 27E

Chapter 5.3, Problem 29E

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The partial fraction decomposition of the function 3 x 2 + 5 x − 13 ( 3 x + 2 ) ( x 2 − 4 x + 4 ) .

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 3x2+5x−13(3x+2)(x2−4x+4).

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Chapter 5 Solutions

The partial fraction decomposition of the function 3 x 2 + 5 x − 13 ( 3 x + 2 ) ( x 2 − 4 x + 4 ) .

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 3x2+5x−13(3x+2)(x2−4x+4).

Want to see the full answer?

Chapter 5 Solutions

To find:

The partial fraction decomposition of the function 3x2+5x−13(3x+2)(x2−4x+4).