Key Concepts

• Decompose [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] by writing the partial fractions as [latex]\frac{A}{{a}_{1}x+{b}_{1}}+\frac{B}{{a}_{2}x+{b}_{2}}[/latex]. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations.
• The decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] with repeated linear factors must account for the factors of the denominator in increasing powers.
• The decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in [latex]\frac{A}{x}+\frac{Bx+C}{\left(a{x}^{2}+bx+c\right)}[/latex].
• In the decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex], where [latex]Q\left(x\right)[/latex] has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

  [latex]\frac{Ax+B}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\cdots \text{+}\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}[/latex].

Glossary

partial fractions

the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression

partial fraction decomposition

the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions