1. Partial Fraction Decomposition
Sec. 7.4a

2. First, remind me……………………..…what’s a rational function?
with
In this section, we will write a rational function as a sum of
rational functions where each denominator is a power of a
linear factor or a power of an irreducible quadratic factor.
Example:
Each fraction in the sum is a partial fraction, and the sum is
a partial fraction decomposition of the original rational
function.

3. Steps to Partial Fraction Decomposition of f(x)/d(x)
1. If the degree of f > degree of d: Divide f by d to obtain the
quotient q and the remainder r and write
2. Factor d(x) into a product of factors of the form or
, where is irreducible

4. Steps to Partial Fraction Decomposition of f(x)/d(x)
3. For each factor : The partial fraction decomposition
of r(x)/d(x) must include the sum
where are real numbers
4. For each factor : The partial fraction decomp.
of r(x)/d(x) must include the sum
where and are real numbers

5. Guided Practice
Write the terms for the partial fraction decomposition of the given
rational function. Do not solve for the corresponding constants. + +
Today, we’ll just focus on the
linear factors, like these…

6. by multiplying everything
Guided Practice
Find the partial fraction decomposition of the given function.
Factor the
denominator!
Write the
partial fractions!
“Clear the fractions”
by multiplying everything
by the denominator!

7. Can we verify this answer algebraically? Graphically?
Guided Practice
Find the partial fraction decomposition of the given function.
Equate the coefficients from
each side of the equation!
Solve the system!
(I don’t care how!!!)
Can we verify this answer algebraically? Graphically?

8. Guided Practice
Find the partial fraction decomposition of the given function.
Another (easier?) way to
solve for the constants:
Plug in 5 for x, then
plug in –3 for x:

9. Expand and combine like terms:
Guided Practice
Find the partial fraction decomposition of the given function.
Clear fractions:
Expand and combine like terms:

10. Compare coefficients:
Guided Practice
Find the partial fraction decomposition of the given function.
Compare coefficients:
Solve the system:

11. Guided Practice
Find the partial fraction decomposition of the given function.
The other way to solve for the A’s:
Use x = 2, solve for A 3
Use x = 0, solve for A 1
Use any other x, solve for A 2

12. More PFD: Denominators with Irreducible Quadratic Factors

13. Now let’s apply a similar process when working with irreducible
quadratic factors…
(see “Step 4” in your notes
from the previous slides!!)

14. Find the partial fraction decomposition of
Factor the denominator by grouping:
Clear fractions:
Expand and combine like terms:

15. Find the partial fraction decomposition of
Compare coefficients:
Solve the system:

16. Find the partial fraction decomposition of
Clear fractions:
Expand and combine like terms:

17. Find the partial fraction decomposition of
Compare coefficients:

18. Find the partial fraction decomposition of
Clear fractions:
Expand and combine like terms:

19. Find the partial fraction decomposition of
Compare coefficients: