Partial Fraction Decomposition

Breaks down a fraction in the form $\frac{P(s)}{Q(s)}$ where $\text{deg}(P)<\text{deg}(Q)$

Cases

Non-repeated Linear Factors

• If $Q(s)=(s-r_1)(s-r_2)\cdots (s-r_n)$ where $r_i$ are distinct real numbers
• Then $\frac{P(s)}{Q(s)}=\frac{A_1}{s-r_1}+\frac{A_2}{s-r_2}+\cdots +\frac{A_n}{s-r_n}$ where $A_i$ are real numbers
• Use the cover up method

Repeated Linear Factors

• If $Q(s)$ contains $(s-r{)}^m$ where $m$ is the highest power of $s-r$ that divides $Q(s)$
• Then the portion that corresponds to $(s-r{)}^m$ is $\frac{A_1}{s-r}+\frac{A_2}{(s-r{)}^r}+\cdots +\frac{A_m}{(s-r{)}^m}$ where $A_i$ are real numbers
• Steps
  1. Cover up method the highest degree term
  2. Plug in method if only one unknown coefficient after cover up, polynomial comparison otherwise

Quadratic Factors

If $Q(s)$ contains the quadratic factor $[(s+\alpha {)}^2+{\beta }^2]$ where $m$ is the highest power of $s-r$ that divides $Q(s)$

Then the portion that corresponds to that is$\frac{A_{1}s+B_1}{(s+\alpha {)}^2+{\beta }^2}+\cdots +\frac{A_{m}s+B_m}{[(s+\alpha {)}^2+{\beta }^2{]}^m}$where $A_i$ are real numbers

Alternate form:

$$\frac{A_1(s+\alpha )+B_1\beta }{(s+\alpha {)}^2+{\beta }^2}+\cdots +\frac{A_m(s+\alpha )+B_m\beta }{[(s+\alpha {)}^2+{\beta }^2{]}^m}$$

Methods

• Cover Up
  1. Set $s$ to be each value that causes the $Q(s)$ to be 0
  2. Cover up that $s-a$ on the original and plug that value of $s$ into the rest
  3. Repeat for each
• Plug In Method
  • Plug in a value of $s$
  • Use 0 whenever possible
• Comparing Polynomails
  1. Multiply everything by the common denominator
  2. Algebra to get one case of each degree of $s$
  3. Set the coefficients of each degree of $s$ across the equal sign equal to each other and solve the system of equations

Notes

• Number of coefficients = degree of $Q(s)$