Download Lesson Plan #6

1
Name:
Class: AP Calculus
Per:
Date: Wednesday May 25th, 2011
Topic: Partial Fraction Decomposition
Aim: How can we find the partial fraction decomposition of a rational expression?
Objectives:
1) Students will be able to find the partial fraction decomposition of a rational expression
HW# 90:
Page 647 #’s 1, 2, 3
Do Now:
1) Evaluate
2x 1
dx
2
 4 x
2) Combine the two fractions below:
2
1

x 3 x  2
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Suppose we want to split up the fraction
x7
A
B

into the form
. (Why would we want to do this?)
x  x6
x3 x2
2
Let’s see how we could do this. Basically we want to solve the equation
x7
A
B


.
x  x6 x3 x2
2
How do we usually solve a fractional equation?
x  7  A( x  2)  B( x  3)
x  7  Ax  2 A  Bx  3B
x  7  Ax  Bx  2 A  3B
A B 1
2 A  3B  7
Solve the system
of equations
Example #1:
Write the Partial Fraction Decomposition for the rational expression
Example #2:
Write the Partial Fraction Decomposition for the rational expression
5
x  x6
2
5 x  10
2 x 2  3x  2
2
Let’s examine this next case:
3
2x  3
.
( x  1)2
In this case, a factor (namely  x  1 ) repeats. In this case, we have to set it in the following way
2x  3
 x  1
2

A
B

x  1  x  12
2 x  3  A( x  1)  B
2x  3  Ax  A  B
2 x  3  Ax  ( A B)
2  A , 3   A  B
Implies that B = -1
Example #3:
Find the partial fraction decomposition of
3x
 x  3
2
Example #4:
Write the partial fraction decomposition of
5 x 2  20 x  6
x  x  1
2
4
Example #5:
Find the partial fraction decomposition of
4 x2  2 x 1
x 2 ( x  1)
Example #6:
Find the partial fraction decomposition of
x2 1
x( x 2  1)
In this case, the denominator contains an irreducible quadratic factor, so we have to set it up like this
x2 1
A Bx  c
  2
2
x x 1
x x 1


Example #7:
Find the partial fraction decomposition of
x 2  3x
2 x  1 x 2  4

