Explanation Calculation: Consider the expression, x 2 + 5 x + 1 x 2 − 2 x + 3 . Note that, the highest power of the variable is the degree of an expression. Thus, the degree of the numerator is 2 and the degree of the denominator is 3 . The given rational expression is a proper fraction because the degree of the numerator is less than the degree of the denominator. The denominator is in the factorised from as x + 1 x 2 − 2 x + 3 . Consider, degree of the factor x + 1 is 1 , which is linear, and that of the factor x 2 − 2 x + 3 is 2 , which is quadratic. Since the factor x + 1 is linear and x 2 − 2 x + 3 is quadratic, write one partial fraction with its numerator as an unknown constant for the linear factor and another partial fraction with an unknown linear numerator for the quadratic factor. Thus, x 2 + 5 x + 1 x 2 − 2 x + 3 = A x + 1 + B x + C x 2 − 2 x + 3 The least common denominator (LCD) of the terms on the right side is x + 1 x 2 − 2 x + 3 . So, in order to make the denominators same, A x + 1 + B x + C x 2 − 2 x + 3 = A x 2 − 2 x + 3 x + 1 x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 = A x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 Thus, x 2 + 5 x + 1 x 2 − 2 x + 3 = A x 2 − 2 x + 3 + B x + C x + 1 x + 1 x 2 − 2 x + 3 . Now, the denominators on both sides are same. So, multiply both sides by the LCD to get the basic equation. x 2 + 5 = A x 2 − 2 x + 3 + B x + C x + 1 To solve the basic equation for the unknowns A , B and C , simplify the basic equation. x 2 + 5 = A x 2 − 2 x + 3 + B x + C x + 1 = A x 2 − 2 A x + 3 A + B x 2 + B x + C x + C = A + B x 2 + − 2 A + B + C x + 3 A + C Equate the like terms on either sides of the last equation to get. 1 = A + B …… (1) 0 = − 2 A + B + C …… (2) 5 = 3 A + C …… (3) Multiply equation (1) by − 1 to get − 1 = − A − B …… (4) Add equations (2), (3) and (4) to get

10th Edition

ISBN: 9781337282291

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.4, Problem 35E

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To calculate: The form of partial fraction decomposition of the rational expression x2+5x+1x2−2x+3 and check the answer algebraically.

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Chapter 6.4, Problem 34E

Chapter 6.4, Problem 36E

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The form of partial fraction decomposition of the rational expression x 2 + 5 x + 1 x 2 − 2 x + 3 and check the answer algebraically.

Solution Summary: The author calculates the form of partial tion decomposition of the rational expression, x2+5(x+1) and check the answer algebraically.

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To calculate: The form of partial fraction decomposition of the rational expression x2+5x+1x2−2x+3 and check the answer algebraically.

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