Section A.12 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.

🔗