Key Concepts

• Decompose $\frac{P\left(x\right)}{Q\left(x\right)}$ by writing the partial fractions as $\frac{A}{{a}_{1}x+{b}_{1}}+\frac{B}{{a}_{2}x+{b}_{2}}$. 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 $\frac{P\left(x\right)}{Q\left(x\right)}$ with repeated linear factors must account for the factors of the denominator in increasing powers.
• The decomposition of $\frac{P\left(x\right)}{Q\left(x\right)}$ with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in $\frac{A}{x}+\frac{Bx+C}{\left(a{x}^{2}+bx+c\right)}$.
• In the decomposition of $\frac{P\left(x\right)}{Q\left(x\right)}$, where $Q\left(x\right)$ 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

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

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