sullivan algebra and trigonometry: section 12.6 objectives of this section decompose p/q, where q...
TRANSCRIPT
Sullivan Algebra and Trigonometry: Section 12.6
Objectives of this Section
• Decompose P/Q, Where Q Has Only Nonrepeated Factors
• Decompose P/Q, Where Q Has Repeated Factors
• Decompose P/Q, where Q Has Only Nonrepeated Irreducible Quadratic Factors
• Decompose P/Q, where Q Has Only Repeated Irreducible Quadratic Factors
Recall the following problem from Chapter R:
22
44x x+
++
= + + ++ +
2 4 4 22 4
( ) ( )( )( )x xx x
= ++ +
6 16
6 82
x
x x
In this section, the process will be reversed, decomposing rational expressions into partial fractions.
A rational expression P / Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
CASE 1: Q has only nonrepeated linear factors.
Under the assumption that Q has only nonrepeated linear factors, the polynomial Q has the form
Q x x a x a x an( ) ( )( ) ( )= − − ⋅ ⋅ −1 2 K
where none of the numbers ai are equal. In this case, the partial fraction decomposition of P / Q is of the form
P xQ x
Ax a
Ax a
Ax a
n
n
( )( )
=−
+−
+ +−
1
1
2
2L
where the numbers Ai are to be determined.
Write the partial fraction decomposition of
5 1
2 152
x
x x
−− −
x x x x2 2 15 5 3− − = − +( )( )
5 1
2 15 5 32
x
x x
Ax
Bx
−− −
=−
++
5 1 3 5x A x B x− = + + −( ) ( )5 1 3 5x A B x A B− = + + −( )
5 1 3 5x A B x A B− = + + −( )A B
A B
+ =− =−
53 5 1
Solve this system using either substitution or elimination.
A B= =3 2,
5 1
2 15
35
232
x
x x x x−
− −=
−+
+
CASE 2: Q has repeated linear factors.
If the polynomial Q has a repeated factor, say, (x - a) n, n > 2 an integer, then, in the partial fraction decomposition of P / Q, we allow for the terms
Ax a
A
x a
A
x an
n1 2
2−+
−+ +
−( ) ( )L
Write the partial fraction decomposition of
− −− +x x
x x x
2
3 24 4x x x x x x x x3 2 2 24 4 4 4 2− + = − + = −( ) ( )
− −− +
= +−
+−
x x
x x x
Ax
Bx
C
x
2
3 2 24 4 2 2( )
Case 1↑678
Case 2
1 244 34 4
− − + = − + − +x x A x Bx x Cx2 28 2 2( ) ( )
− − + = − + − +x x A x Bx x Cx2 28 2 2( ) ( )− − + = − + + − +x x A x x Bx Bx Cx2 2 28 4 4 2( )− − + = + + − − + +x x A B x A B C x A2 28 4 2 4( ) ( )
Solve the matrix system using Cramer’s Rule
1 1 0 1
4 2 1 1
4 0 0 8
−− − −⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
A B C= =− =2 3 1, ,
− −− +
= +−
+−
x x
x x x
Ax
Bx
C
x
2
3 2 24 4 2 2( )
A B C= =− =2 3 1, ,
− −− +
= + −−
+−
x x
x x x x x x
2
3 2 24 4
2 32
1
2( )
CASE 3: Q contains a nonrepeated irreducible quadratic factor.
If Q contains a nonrepeated irreducible quadratic factor of the form ax2 + bx + c, then, in the partial fraction decomposition of P / Q, allow for the term
Ax B
ax bx c
++ +2
where the numbers A and B are to be determined.
Write the partial fraction decomposition of
4 3 5
1
3 2
4
x x
x
− −−
x x x x x x4 2 2 21 1 1 1 1 1− = − + = + − +( )( ) ( )( )( )
4 3 5
1 1 1 1
3 2
4 2
x x
x
Ax
Bx
Cx D
x
− −−
=+
+−
+ ++
4 3 5
1 1 1 1 1
3 2
2 2 2
x x
A x x B x x Cx D x
− −
= − + + + + + + −( )( ) ( )( ) ( )( )
4 3 5
1 1 1 1 1
3 2
2 2 2
x x
A x x B x x Cx D x
− −
= − + + + + + + −( )( ) ( )( ) ( )( )
Let x = 1:
-4 = 4B
B = -1
Let x = -1: -12 = -4A
A = 3
4 3 5
3 1 1 1 1 1
1
3 2
2 2
2
x x
x x x x
Cx D x
− −
= − + + − + +
+ + −
( )( ) ( )( )( )
( )( )
4 3 5 3 3 3 3 1
1
3 2 3 2 3 2
2
x x x x x x x x
Cx D x
− − = − + − − − − −
+ + − ( )( )
2 2 1 13 2 2x x x Cx D x+ − − = + −( )( )x x x Cx D x2 22 1 1 2 1 1( ) ( ) ( )( )+ − + = + −
( )( ) ( )( )2 1 1 12 2x x Cx D x+ − = + −
( ) ( )2 1x Cx D+ = +
C D= =2 1,
4 3 5
1 1 1 1
3 2
4 2
x x
x
Ax
Bx
Cx D
x
− −−
=+
+−
+ ++
A B C D= =− = =3 1 2 1, , ,
4 3 5
1
31
11
2 1
1
3 2
4 2
x x
x x xx
x
− −−
=+
+ −−
+ ++
CASE 4: Q contains repeated irreducible quadratic factors.
If the polynomial Q contains a repeated irreducible quadratic factor (ax2 + bx + c)n, n > 2, n an integer, then, in the partial fraction decomposition of P / Q, allow for the terms
( ) ( )A x B
ax bx c
A x B
ax bx c
A x B
ax bx c
n nn
1 12
2 2
2 2 2
+
+ ++
+
+ ++ +
+
+ +L
where the numbers Ai ,Bi are to be determined.
( )
Write the partial fraction decomposition of:
x x
x
2
2 29
+
+
( ) ( )x x
x
Ax B
x
Cx D
x
2
2 2 2 2 29 9 9
+
+=
+
++
+
+
x x Ax B x Cx D2 2 9+ = + + + +( )( )
x x Ax Bx A C x B D2 3 2 9 9+ = + + + + +( )
x x Ax Bx A C x B D2 3 2 9 9+ = + + + + +( )A
B
A C
B D
==+ =+ =
01
9 19 0
A B C D= = = =−0 1 1 9, , ,
( ) ( )x x
x x
x
x
2
2 2 2 2 29
1
9
9
9
+
+=
++
−
+