Section A.6 Partial Fraction Decomposition

Check the Degree of the Numerator and Denominator

Ensure the degree of the numerator is less than the degree of the denominator. If the numerator has a degree greater than or equal to the denominator, first perform polynomial long division to reduce the degree of the numerator.

Factor the Denominator

Factor the denominator into irreducible linear or quadratic factors.

• Linear factors: Expressions of the form \((s - a)\text{.}\)
• Irreducible quadratic factors: Expressions of the form \((s^2 + bs + c)\) where the discriminant \(b^2 - 4ac\) is negative.

Set up the Partial Fraction Decomposition

Based on the factors of the denominator, write the decomposition:

• For each linear factor \((s - a)\text{,}\) include a term of the form

  \begin{equation*} \frac{A}{s - a}\text{.} \end{equation*}
• For repeated linear factors \((s - a)^n\text{,}\) include terms like:

  \begin{equation*} \frac{A_1}{s - a} + \frac{A_2}{(s - a)^2} + \dots + \frac{A_n}{(s - a)^n}. \end{equation*}
• For irreducible quadratic factors \((s^2 + bs + c)\text{,}\) include a term of the form

  \begin{equation*} \frac{Bs + C}{s^2 + bs + c}\text{.} \end{equation*}
• For repeated quadratic factors \((s^2 + bs + c)^n\text{,}\) include terms like:

  \begin{equation*} \frac{B_1s + C_1}{s^2 + bs + c} + \frac{B_2s + C_2}{(s^2 + bs + c)^2} + \dots + \frac{B_ns + C_n}{(s^2 + bs + c)^n}. \end{equation*}

Solve for the Constants

Multiply both sides of the equation by the common denominator and expand to eliminate fractions. Group terms by powers of \(s\) and set up a system of equations to solve for the unknown constants \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\)

Final Result

Once the constants are found, write the final partial fraction decomposition. This decomposition can now be used for further calculations such as integrals or inverse Laplace transforms.