Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.
(a)
\(\frac{{{x^4} - 2{x^3} + {x^2} + 2x - 1}}{{{x^2} - 2x + 1}}\)
(b)
\(\frac{{{x^2} - 1}}{{{x^3} + {x^2} + x}}\)

\[\frac{{{x^2}}}{{{x^2} - 2x + 1}} + \frac{{2x}}{{{x^2} - 2x + 1}} - \frac{1}{{{x^2} - 2x + 1}} + \frac{{{x^4}}}{{{x^2} - 2x + 1}} - \frac{{2{u^3}}}{{{x^2} - 2x + 4}}\]

This is the final answer.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Decomposition of the partial fraction.

\(\frac{{{x^4} - 2{x^3} + {x^2} + 2x - 1}}{{{x^2} - 2x + 1}}\)\(\)

\({x^2} - 2x + 1 = {\left( {x - 1} \right)^2}\)

Step 2: Calculation of partial fraction.

\({x^4} - 2{x^3} + {x^2} + 2x - 1 = {x^2}\left( {{x^2} - 2x + 1} \right) + (2x - 1)\)

Written as

\({x^2} + \frac{{2x - 1}}{{{{(x - 1)}^2}}} = \frac{{x(x(x - 2)(x + 1) + 2) - 1}}{{(x - 2)x + 1}}\)

Expansion

\[\frac{{{x^2}}}{{{x^2} - 2x + 1}} + \frac{{2x}}{{{x^2} - 2x + 1}} - \frac{1}{{{x^2} - 2x + 1}} + \frac{{{x^4}}}{{{x^2} - 2x + 1}} - \frac{{2{u^3}}}{{{x^2} - 2x + 4}}\]

Hence, This is the final answer.

(b) Step 1:

\({x^3} + {x^2} + x = x\left( {{x^2} + x + 1} \right)\)

(b) Step 2: Calculation of Partial fraction.

\(\frac{{{x^2} - 1}}{{{x^3} + {x^2} + x}} = \frac{A}{x} + \frac{{Bx + C}}{{{x^2} + x + 1}}\)

Hence \(\frac{A}{x} + \frac{{Bx + c}}{{{x^2}(x + 1)}}\) is final answer.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.
(a)
\(\frac{{{x^4} - 2{x^3} + {x^2} + 2x - 1}}{{{x^2} - 2x + 1}}\)
(b)
\(\frac{{{x^2} - 1}}{{{x^3} + {x^2} + x}}\)

Step by Step Solution

Step 1: Decomposition of the partial fraction.

Step 2: Calculation of partial fraction.

(b) Step 1:

(b) Step 2: Calculation of Partial fraction.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.(a) \(\frac{{{x^4} - 2{x^3} + {x^2} + 2x - 1}}{{{x^2} - 2x + 1}}\)(b)\(\frac{{{x^2} - 1}}{{{x^3} + {x^2} + x}}\)

Step by Step Solution

Step 1: Decomposition of the partial fraction.

Step 2: Calculation of partial fraction.

(b) Step 1:

(b) Step 2: Calculation of Partial fraction.

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Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.
(a)
\(\frac{{{x^4} - 2{x^3} + {x^2} + 2x - 1}}{{{x^2} - 2x + 1}}\)
(b)
\(\frac{{{x^2} - 1}}{{{x^3} + {x^2} + x}}\)