Explanation Given information: The function is 7 x − 3 x 3 + 2 x 2 − 3 x . Concept used: Consider p ( x ) be any rational function that can be expressed as, p ( x ) = P ( x ) Q ( x ) Consider that Q ( x ) factors as, Q ( x ) = ( m 1 x + n 1 ) ( m 2 x + n 2 ) ( m 3 x + n 3 ) ⋯ ( m k x + n k ) The factors of Q ( x ) are not repeated. Then partial fraction decomposition of the function p ( x ) can be expressed as, P ( x ) Q ( x ) = A 1 m 1 x + n 1 + A 2 m 2 x + n 2 + A 3 m 3 x + n 3 + ⋯ A k m k x + n k Here, A 1 , A 2 , A 3 , … A k are constants. Calculation: The given function is 7 x − 3 x 3 + 2 x 2 − 3 x . Compare the above function with the form P ( x ) Q ( x ) to obtain the value of P ( x ) and Q ( x ) as, P ( x ) = 7 x − 3 Q ( x ) = x 3 + 2 x 2 − 3 x The denominator Q ( x ) is x 3 + 2 x 2 − 3 x . The factorization of the denominator Q ( x ) can be calculated as, Q ( x ) = x 3 + 2 x 2 − 3 x = x ( x 2 + 2 x − 3 ) = x ( x 2 + 3 x − x − 3 ) = x ( x ( x + 3 ) − ( x + 3 ) ) Further simplify the above equation of Q ( x ) as, Q ( x ) = x ( x + 3 ) ( x − 1 ) The factors of Q ( x ) are x , ( x + 3 ) and ( x − 1 ) . Therefore, the partial fraction decomposition of the function is, 7 x − 3 x 3 + 2 x 2 − 3 x = A x + B x + 3 + C x − 1 …… (1) Here, A , B and C are constants. The common denominator of x , ( x + 3 ) and ( x − 1 ) is x ( x + 3 ) ( x − 1 )

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 5.3, Problem 24E

To determine

To find:

The partial fraction decomposition of the function 7x−3x3+2x2−3x.

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Chapter 5.3, Problem 23E

Chapter 5.3, Problem 25E

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