To explain: the partial fraction decomposition of a rational expression.
Answer to Problem 17RCC
The partial fraction decomposition of a rational expression is explained.
Explanation of Solution
Consider a rational function
factoring its denominator and setting it equal to a sum of fractions using the terms from the denominator. These fractions will have undetermined coefficients. There are four cases to consider.
Case 1: The denominator only has distinct linear terms.
If the denominator of
Where A, are undetermined coefficients and
Case 2: there will be repeated linear factors in the denominator
If the denominator of
Where A, are the undetermined coefficients and
Case 3: The denominator contains distinct irreducible quadratic factors
If the denominator of
Where A and B are the undetermined coefficients and
Case 4: There will be repeated irreducible quadratic factors in the denominator
If the denominator of
Where the
Once a sum of fractions with undetermined coefficients is obtained, cross multiply and collect like terms.
In order to solve for the undetermined coefficients, look at both sides of the obtained equation and set the coefficients for different powers of x equal to each other.
This gives us a linear system of equations involving the different coefficients.
Once the system of equations is solved, substitute the values that are obtained back into the
sum of fractions. This sum of fractions is the partial fraction decomposition of
Chapter 10 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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