5017 Partial Fractions

AP Calculus

Partial Fractions

 

Remember: Adding Fractions Butterfly Method:    

 

Decomposition: GIVEN: SUM FIND: ADDENDS ( Like Factoring- given the product (answer) want to find the problem) 

a c

b d

Steps:• ** Check if proper fraction• Factor the denominator and write the addition problem.

Write fraction addition problem using a numerator one degree less than the denominator; may have repeated factor. If

a. Linear factors, use

b. Quadratic factors, use

3. Add the new fraction; find LCD, etc.4. Set numerators equal.5. Solve

2

Ax B

ax bx c

A

mx b

a. Convenient x-values.b. Equate the coefficients.

Examples: Linear factors

Example 1: Linear factors

2

5 16

2 11 5

x

x x

Examples: Linear factors

Example 2: Linear factors

2

7

6

xdx

x x

Logistic Growth

( )dy

ky L ydt

Examples: Repeated Linear factors

Example 3: Repeated Linear Factors 

Must do all possibilities up to the given exponent.

3 2 3( 2) ( 2) ( 2) ( 2)

A B C

x x x x

2

3 2

5 20 6

2

x x

x x x

Examples: Repeated Linear factors

Example 4: Repeated Linear Factors2

3

4 3

( 5)

xdx

x

Examples: Quadratic Factors

2 2( 5)( 4) ( 5) ( 4)

A Bx C

x x x x

Example 5: 2

3 2

4 4

2 2

x x

x x x

Examples: Quadratic Factors

Example 6: 2( 1)( 1)

xdx

x x x

Examples: Quadratic Factors

2 2( 5)( 4) ( 5) ( 4)

A Bx C

x x x x

Example 7:2

2 2

2 8

( 4)

x x

x

Last Update:

• 03/08/10