Explanation Given: We can factor the denominator of a rational function into distinct linear factors. D ( x ) = ( x − c 1 ) ( x − c 2 ) ... ( x − c n ) Where c 1 , c 2 , ... c n are distinct linear factors. Here, P k = N ( c k ) D ′ ( c k ) where N is a polynomial degree less than n and k = 1 , 2 , 3 , ... , n , which is partial fraction decomposition of N ( x ) D ( x ) . Formula used: Product Rule of Differentiation states that for functions u and v d d x ( u ⋅ v ) = u ′ v + v ′ u Proof: Assume N ( x ) D ( x ) = P 1 x − c 1 + P 2 x − c 2 + ... + P n x − c n to be true. Then take LCM N ( x ) ( x − c 1 ) ( x − c 2 ) ... ( x − c n ) = { P 1 ( x − c 2 ) ( x − c 3 ) ... ( x − c n ) + P 2 ( x − c 1 ) ( x − c 3 ) ... ( x − c n ) + ... + P n ( x − c 1 ) ( x − c 2 ) ... ( x − c n − 1 ) } ( x − c 1 ) ( x − c 2 ) ... ( x − c n ) N ( x ) = { P 1 ( x − c 2 ) ( x − c 3 ) ... ( x − c n ) + P 2 ( x − c 1 ) ( x − c 3 ) ... ( x − c n ) + ... + P n ( x − c 1 ) ( x − c 2 ) ... ( x − c n − 1 ) } Let x = c 1 . Therefore, N ( c 1 ) = P 1 ( c 1 − c 2 ) ( c 1 − c 3 ) ... ( c 1 − c n ) P 1 = N ( c 1 ) ( c 1 − c 2 ) ( c 1 − c 3 ) ... ( c 1 − c n ) Similarly, let x = c 2 N ( c 2 ) = P 2 ( c 2 − c 1 ) ( c 2 − c 3 ) ... ( c 2 − c n ) P 2 = N ( c 2 ) ( c 2 − c 1 ) ( c 2 − c 3 )

Preparing For The Ap Calculs Ab & Calculus Bc Examinations To Accompany Calculus And Calculus Of A Single Variable 10th & 11th Editions

11th Edition

ISBN: 9781337286961

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 8, Problem 11PS

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To prove: The relation N(x)D(x)=P1x−c1+P2x−c2+...+Pnx−cn where D(x)=(x−c1)(x−c2)...(x−cn)

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Chapter 8, Problem 10PS

Chapter 8, Problem 12PS

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Chapter 8 Solutions

Preparing For The Ap Calculs Ab & Calculus Bc Examinations To Accompany Calculus And Calculus Of A Single Variable 10th & 11th Editions

Ch. 8.1 - Integration Technique Describe how to integrate a...Ch. 8.1 - Fitting Integrands to Basic Integration Rules What...Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...

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To prove: The relation N ( x ) D ( x ) = P 1 x − c 1 + P 2 x − c 2 + ... + P n x − c n where D ( x ) = ( x − c 1 ) ( x − c 2 ) ... ( x − c n )

Solution Summary: The author explains that we can factor the denominator of a rational function into distinct linear factors.

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To prove: The relation N(x)D(x)=P1x−c1+P2x−c2+...+Pnx−cn where D(x)=(x−c1)(x−c2)...(x−cn)

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Chapter 8 Solutions