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 cb 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 Bax bx c
Amx b
a. Convenient x-values.b. Equate the coefficients.
Examples: Linear factors
Example 1: Linear factors2
5 162 11 5
xx x
Examples: Linear factors
Example 2: Linear factors
2
76
x dxx 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 62
x xx x x
Examples: Repeated Linear factors
Example 4: Repeated Linear Factors2
3
4 3( 5)x dxx
Examples: Quadratic Factors
2 2( 5)( 4) ( 5) ( 4)A Bx C
x x x x
Example 5: 2
3 2
4 42 2
x xx x x
Examples: Quadratic Factors
Example 6: 2( 1)( 1)x dx
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 xx
Last Update:
• 03/08/10
