Explanation Given information: The rational expression is 5 x 2 − 3 x + 10 x 4 + x 2 − 2 . Concept used: Concept used: Irreducible Quadratic Factors: Consider p ( x ) be any ration function that can be expressed as, p ( x ) = P ( x ) Q ( x ) Also consider that if Q ( x ) has quadratic factor ( m x 2 + n x + k ) and the factor ( m x 2 + n x + k ) cannot be factored further. Then partial fraction decomposition of the rational function p ( x ) can be expressed as, P ( x ) Q ( x ) = A x + B m x 2 + n x + k Here, A and B are constants. Calculation: The given rational expression is 5 x 2 − 3 x + 10 x 4 + x 2 − 2 . Compare the expression with the form P ( x ) Q ( x ) to obtain the value of P ( x ) and Q ( x ) as, P ( x ) = 5 x 2 − 3 x + 10 Q ( x ) = x 4 + x 2 − 2 The denominator Q ( x ) of the expression is x 4 + x 2 − 2 . The factorization of the denominator Q ( x ) can be calculated as, Q ( x ) = x 4 + x 2 − 2 = x 4 + 2 x 2 − x 2 − 2 = x 2 ( x 2 + 2 ) − ( x 2 + 2 ) Simplify the above equation of denominator Q ( x ) as, Q ( x ) = ( x 2 − 1 ) ( x 2 + 2 ) = ( x − 1 ) ( x + 1 ) ( x 2 + 2 ) The factors of the denominator Q ( x ) are ( x − 1 ) , ( x + 1 ) and ( x 2 + 2 ) . The factor ( x 2 + 2 ) cannot be factored further. Therefore, the partial fraction decomposition of the function is, 5 x 2 − 3 x + 10 x 4 + x 2 − 2 = A x − 1 + B x + 1 + C x + D x 2 + 2 ......(1) Here, A , B , C and D are constants. The common denominator of ( x − 1 ) , ( x + 1 ) and ( x 2 + 2 ) is ( x − 1 ) ( x + 1 ) ( x 2 + 2 )

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 5, Problem 34E

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The partial fraction decomposition of the rational expression 5x2−3x+10x4+x2−2.

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Chapter 5, Problem 33E

Chapter 5, Problem 35E

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The partial fraction decomposition of the rational expression 5 x 2 − 3 x + 10 x 4 + x 2 − 2 .

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

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The partial fraction decomposition of the rational expression 5x2−3x+10x4+x2−2.

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The partial fraction decomposition of the rational expression 5 x 2 − 3 x + 10 x 4 + x 2 − 2 .

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

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To find: The partial fraction decomposition of the rational expression 5x2−3x+10x4+x2−2.

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Chapter 5 Solutions

To find:

The partial fraction decomposition of the rational expression 5x2−3x+10x4+x2−2.