1. 8.5 Partial Fractions (part 1)
The Empire Builder, 1957
Greg Kelly, Hanford High School, Richland, Washington

2. Objectives Understand the concept of a partial fraction decomposition.
Use partial fractions decomposition with linear factors to integrate rational functions.
Use partial fraction decomposition with quadratic factors to integrate rational functions.

3. Method of partial fractions: decomposing a rational function into simpler rational functions
We could reverse the process and find

4. You can solve
without partial fractions, but it’s messy. You would complete the square and use trig substitution. Part of this solution is on page 515 of the text.
BUT if you know that
then you get

5. Every polynomial with real coefficients can be factored into linear and irreducible quadratic factors.
For example:
linear
factor
irreducible
quadratic
factor
repeated linear
factor

6. Since
we know

7. Rules for Decomposition of N(x)/D(x) into Partial Fractions:
1. Divide if improper (if degree of numerator ≥ degree of denominator).
2. Factor denominator completely.
3. Linear factors: For each factor of the form (px+q)m, the partial fraction decomposition must include
4. Quadratic Factors: For each factor of the form
(ax2 + bx + c)n, the decomposition must include

8. Distinct Linear Factors:
Factor completely.
These are called non-repeating linear factors.

9. The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside.
Multiply by the common denominator.
Solve for A: let x=3
Solve for B: let x=2

11. Multiply by the common denominator.
Let x = - 1
Let x = 3

13. Repeated Linear Functions:
Repeated roots: we must use two terms for partial fractions.

14. Let x=0:
Let x= -1:
“Convenient” values have been used, so let x=1:

16. If the degree of the numerator is higher than the degree of the denominator, use long division first.
(from example one)

17. Guidelines for Solving the Basic Equation (p. 521)
Linear Factors
Substitute for roots of the distinct linear factors into the basic equation.
For repeated linear factors, use the coefficients determined in guideline 1 to rewrite the basic equation. Then substitute other convenient values of x and solve for the remaining coefficients.

18. Homework
8.5 (page 561)
#1 – 17 odd