Explanation Given: 1 x 3 − 1 Concept Used: To convert a complicated single fraction in partial fraction decomposition we follow the steps Factor the denominator completely into linear factors of the form ( a x + b ) m and quadratic factors of the form ( a x 2 + b x + c ) n that are not further factorable over the integers Set up the decomposition. That is, write the original rational expression as a sum of simpler fraction, as A 1 ( a x + b ) 1 + A 2 ( a x + b ) 2 + ......... + A m ( a x + b ) m , where A 1 , A 2 .... A m are constants And then solve to find the value of these constants For a first-degree (linear) denominator, the numerator is constant (degree 0) For a second-degree (linear) denominator, the numerator is linear (degree 1) Calculation: To set up partial fraction form for 1 x 3 − 1 First factorize the denominator, as x 3 − 1 = ( x − 1 ) ( x 2 + x + 1 ) Now, break up the denominator into parts each with a linear denominator, as 1 x 3 − 1 = 1 ( x − 1 ) ( x 2 + x + 1 ) = A x − 1 + B x + C x 2 + x + 1 Now we have to solve value of A, B, and C, as 1 ( x − 1 ) ( x 2 + x + 1 ) = A x − 1 + B x + C x 2 + x + 1 = A ( x 2 + x + 1 ) + ( B x + C ) ( x − 1 ) ( x − 1 ) ( x 2 + x + 1 ) = A x 2 + A x + A + B x

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 7.4, Problem 10PS

To determine

Find the partial fraction decomposition for

1x3−1.

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Chapter 7.4, Problem 9PS

Chapter 7.4, Problem 11PS

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Chapter 7 Solutions

Calculus

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