Concept explainers
In Problems 47-54, use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition.
To find:
i. Rewrite improper rational expression as the sum of polynomial and a proper rational expression using the division algorithm.
ii. The partial fraction decomposition of the proper rational expression
iii. To express the improper rational expression as the sum of polynomial and the partial fraction decomposition.
Answer to Problem 54AYU
Solution:
Explanation of Solution
Given:
Formula used:
a. If has repeated linear factor of the form an integer, then, in the partial fraction decomposition of , takes the form:
where, the numbers and are to be determined.
b. If has repeated linear factor of the form an integer, then, in the partial fraction decomposition of , takes the form:
where the numbers are to be determined.
Calculation
Divisional Algorithm
From the above division,
----- Eq (1)
To find the partial fraction expression of
The denominator can written be factored as
----- Eq (2)
----- Eq (3)
Equating the co-efficients on both sides in Eq(3),
----- Eq(4)
----- Eq(5)
----- Eq6)
----- Eq(7)
Solving for and from Eq(4) to Eq(7) we get
.
The partial fraction decomposition of is
Finally,
Chapter 11 Solutions
Precalculus Enhanced with Graphing Utilities
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Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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