### Learning Objectives

By the end of this section, you will be able to:

- Decompose [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] , where Q( x ) has only nonrepeated linear factors.
- Decompose [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] , where Q( x ) has repeated linear factors.
- Decompose [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] , where Q( x ) has a nonrepeated irreducible quadratic factor.
- Decompose [latex]\frac{{P( x )}}{{ Q( x )}}[/latex] , where Q( x ) has a repeated irreducible quadratic factor.

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.

Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.