Chapter 10, Problem 17RCC

To determine

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 $r\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ , obtain its partial fraction decomposition by

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 $r\left(x\right)$ only has distinct linear terms, then its partial fraction decomposition will be of the form:

  $r\left(x\right)=\frac{A_1}{\left(a_{1}x+b_1\right)}+\frac{A_1}{\left(a_{2}x+b_2\right)}+\mathrm{...}+\frac{A_n}{\left(a_{n}x+b_n\right)}$

Where A, are undetermined coefficients and $\left(a_{i}x+b_i\right)$ are the n distinct linear factors of the denominator.

Case 2: there will be repeated linear factors in the denominator

If the denominator of $r\left(x\right)$ contains a linear term repeated k times, then its partial fraction decomposition will contain terms of the form:

  $\frac{A_1}{\left(ax+b\right)}+\frac{A_2}{{\left(ax+b\right)}^2}+\mathrm{...}+\frac{A_k}{{\left(ax+b\right)}^k}$

Where A, are the undetermined coefficients and $\left(ax+b\right)$ is the linear factor repeated in the denominator of $r\left(x\right)$ .

Case 3: The denominator contains distinct irreducible quadratic factors

If the denominator of $r\left(x\right)$ contains a distinct irreducible quadratic factor of the form $a{x}^2+bx+c$ then its partial fraction decomposition will contain the term of the form:

  $\frac{Ax+b}{\left(a{x}^2+bx+c\right)}$

Where A and B are the undetermined coefficients and $\left(a{x}^2+bx+c\right)$ is the linear factor repeated in the denominator of $r\left(x\right)$ .

Case 4: There will be repeated irreducible quadratic factors in the denominator

If the denominator of $r\left(x\right)$ contains a quadratic term repeated k times, then its partial fraction decomposition will contain terms of the form:

  $\frac{A_{1}x+B_1}{\left(a{x}^2+bx+c\right)}+\frac{A_{2}x+B_2^{}}{{\left(a{x}^2+bx+c\right)}^2}+\mathrm{...}+\frac{A_{k}x+B_k}{{\left(a{x}^2+bx+c\right)}^k}$

Where the $A_i$ and $B_i$ are undetermined coefficients and $\left(a{x}^2+bx+c\right)$ is the irreducible quadratic factor repeated in the denominator of $r\left(x\right)$ .

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 $r\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ .