First, remind me……………………..…what’s a rational function?
f x
y xg x
0g x with
In this section, we will write a rational function as a sum ofrational functions where each denominator is a power of alinear factor or a power of an irreducible quadratic factor.
Example:2
3 4
2
x
x x
2 1
2x x
Each fraction in the sum is a partial fraction, and the sum isa partial fraction decomposition of the original rationalfunction.
Steps to Partial Fraction Decomposition of f(x)/d(x)
f x r x
q xd x d x
umx n
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 2 vax bx c 2 v
ax bx c
Steps to Partial Fraction Decomposition of f(x)/d(x)
1 2
2u
u
AA A
mx n mx n mx n
umx n3. For each factor : The partial fraction decomposition of r(x)/d(x) must include the sum
where are real numbers1 2, , , uA A A
1 1 2 2
22 2 2
v vv
B x CB x C B x C
ax bx c ax bx c ax bx c
2 vax bx c 4. For each factor : The partial fraction decomp.
of r(x)/d(x) must include the sum
where and are real numbers1 2, , , VB B B1 2, , , vC C C
Guided PracticeGuided Practice
3 2
5 1
3 1
x
x x x
Write the terms for the partial fraction decomposition of the givenrational function. Do not solve for the corresponding constants.
31 22 3
AA A
x x x 4
3
A
x 1 12 1
B x C
x
+ +
Today, we’ll just focus on thelinear factors, like these…
Guided PracticeGuided Practice
2
5 1
2 15
x
x x
Find the partial fraction decomposition of the given function.
5 1
3 5
x
x x
Factor thedenominator!
1 25 1
3 5 3 5
A Ax
x x x x
Write the
partial fractions!
1 25 1 5 3x A x A x “Clear the fractions”by killing the denominator!
1 1 2 25 1 5 3x A x A A x A
1 2 1 25 1 5 3x A A x A A
Guided PracticeGuided Practice
2
5 1
2 15
x
x x
Find the partial fraction decomposition of the given function.
1 2 5A A Equate the coefficients fromeach side of the equation!
1 25 3 1A A
1 2 1 25 1 5 3x A A x A A
Solve the system!(I don’t care how!!!) 1 2A 2 3A
2 3
3 5x x
Can we verify this answer algebraically? Graphically?Can we verify this answer algebraically? Graphically?
Guided PracticeGuided Practice
2
5 1
2 15
x
x x
Find the partial fraction decomposition of the given function.
Another (easier?) way tosolve for the constants:
1 2A 2 3A
2 3
3 5x x
1 25 1 5 3x A x A x
Substitute in 5 for x, thensubstitute in –3 for x:
(these values come from thedenominator, x=-3, and 5)
Guided PracticeGuided Practice
2
3 2
2 4
4 4
x x
x x x
Find the partial fraction decomposition of the given function.
Clear fractions:
221 2 32 4 2 2x x A x A x x A x
2
2
2 4
2
x x
x x
31 2
22 2
AA A
x x x
Expand and combine like terms:
2 21 2 1 2 3 12 4 4 2 4x x A A x A A A x A
Guided PracticeGuided Practice
2
3 2
2 4
4 4
x x
x x x
Find the partial fraction decomposition of the given function.
Compare coefficients:
21 2 2
2 2x x x
1 2 1A A
1 2 34 2 2A A A
14 4A
2 21 2 1 2 3 12 4 4 2 4x x A A x A A A x A
Solve the system: 3 2A 1 1A 2 2A
Guided PracticeGuided Practice
221 2 32 4 2 2x x A x A x x A x
2
3 2
2 4
4 4
x x
x x x
Find the partial fraction decomposition of the given function.
The other way to solve for the A’s:
21 2 2
2 2x x x
3 2A 1 1A 2 2A Use x = 2, solve for A 3
Use x = 0, solve for A 1
Use any other x, solve for A2
Now let’s apply a similar processwhen working with irreducible
quadratic factors…(see “Step 4” in your notesfrom the previous slides!!)
Find the partial fraction decomposition of
2
3 2
4 1
1
x x
x x x
Factor the denominator by grouping:3 2 1x x x 2 1 1x x x
21 1
A Bx C
x x
21 1x x Clear fractions:
2 24 1 1 1x x A x Bx C x Expand and combine like terms:
2 24 1x x A B x B C x A C
Find the partial fraction decomposition of
2
3 2
4 1
1
x x
x x x
2
3 2 2
1 1
x
x x
Compare coefficients: 1A B 2 24 1x x A B x B C x A C
4B C 1A C
Solve the system: 3A 2B 2C
Find the partial fraction decomposition of
3 2
22
2 5
1
x x x
x
1 1 2 2
22 21 1
B x C B x C
x x
Clear fractions:
3 2 21 1 2 22 5 1x x x B x C x B x C
Expand and combine like terms:
3 2 3 21 1 1 2 1 22 5x x x B x C x B B x C C
Find the partial fraction decomposition of
3 2
22
2 5
1
x x x
x
22 2
2 1 3 1
1 1
x x
x x
Compare coefficients:
3 2 3 21 1 1 2 1 22 5x x x B x C x B B x C C
1 2B 1 1C
1 2 5B B 1 2 0C C
2 3B 2 1C
Find the partial fraction decomposition of
2
22
3 4
1
x
x
1 1 2 2
22 21 1
B x C B x C
x x
Clear fractions:
2 21 1 2 23 4 1x B x C x B x C
Expand and combine like terms:
2 3 21 1 1 2 1 23 4x B x C x B B x C C