Avon High School
Section: 8.3
ACE COLLEGE ALGEBRA II - NOTES
Partial Fractions
Mr. Record: Room ALC-129
Semester 2 - Day 3
The Idea Behind Partial Fraction Decomposition
It’s easy to follow that….
But, what if we turned things around….
3
2
3( x  2)  2( x  4)


x4 x2
( x  4)( x  2)
3x  6  2 x  8

( x  4)( x  2)
x  14

.
( x  4)( x  2)
x  14
( x  4)( x  2)
is expressed as the
sum of two simpler
fractions
Partial fraction
Partial fraction
x  14
3
2


( x  4)( x  2) x  4 x  2
This is the partial fraction
decomposition of
These types of problems fall into four different categories.
1.
2.
3.
4.
x  14
( x  4)( x  2)
The denominator is a product of distinct linear factors.
The denominator is a product of linear factors, some of which are repeated.
The denominator has prime quadratic factors, none of which is repeated.
The denominator has a repeated prime quadratic factor.
Example 1
Partial Fraction Decomposition with Distinct Linear Factors
Find the partial fraction decomposition of
5x  1
2
x  x  12
Let
be positive real numbers with
, and let p be any real number.
The logarithm of a number with an exponent is the product of the exponent and
the logarithm
of that number.
Steps in Partial Fraction Decomposition
1. Set up the partial fraction decomposition with the unknown constants, A, B, C, etc, in the
numerator of the decomposition.
2. Multiply both sides of the resulting equation by the least common denominator.
3. Simplify the right-hand side of the equation.
4. Write both sides in descending powers, equate coefficients of like powers of x, and equate
constant terms.
5. Solve the resulting system for A, B, C, etc.
6. Substitute the values for A, B, C, etc., into the equation in Step 1 and write the partial
fraction decomposition.
Example 2
Partial Fraction Decomposition with Repeated Linear Factors
Find the partial fraction decomposition of
x2
x( x  1)2
Example 3
Partial Fraction Decomposition with a Quadratic Factor
Find the partial fraction decomposition of
8 x 2  12 x  20
( x  3)( x 2  x  2)
Example 4
Partial Fraction Decomposition with a Repeated Quadratic Factor
Find the partial fraction decomposition of
2 x3  x  3
( x 2  1)2