1. 8.2: Partial Fractions Rational function Improper rational function
A ratio of two polynomials
Improper rational function
The degree of P is greater than or equal to the degree of Q.
Proper rational function
The degree of Q is greater than the degree of P.

2. Improper Rational Functions
Can be written as the sum of a polynomial and a proper rational function.
Where deg(P(x))  deg(Q(x)) and deg(r(x)) < deg(Q(x))
Use long division of Q(x) into P(x) to accomplish this.

3. Examples

4. Examples

5. Partial Fractions Decomposition
A method for rewriting a proper rational function as a sum of simpler rational functions.
Let’s start with the proper rational function R(x) = P(x)/Q(x).
We need to consider 4 cases…

6. Case 1 Q(x) factors into n linear factors.
In this case, the partial fractions decomposition of R(x) is…

7. Case 2 Q(x) has only linear factors, including some repeated factors.
Suppose Q(x) contains the factor (xa) a total of n times
i.e., Q(x) contains the factor (xa)n
The partial fractions decomposition of R(x) must include:

8. Case 3 Q(x) contains an irreducible quadratic polynomial.
ax2 + bx + c, where b2  4ac < 0.
Has no real roots.
The partial fractions decomposition of R(x) must include:

9. Case 4
Q(x) contains an irreducible quadratic polynomial, raised to the nth power.
(ax2 + bx + c)n, where b2  4ac < 0.
The partial fractions decomposition of R(x) must include: