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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

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Page 1: 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

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

Page 2: 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

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.

Page 3: 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

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.

Page 4: 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

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.

Page 5: 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

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− = + + −( )

Page 6: 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

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−

− −=

−+

+

Page 7: 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

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

Page 8: 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

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( ) ( )

Page 9: 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

− − + = − + − +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, ,

Page 10: 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

− −− +

= +−

+−

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( )

Page 11: 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

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.

Page 12: 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

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

− −

= − + + + + + + −( )( ) ( )( ) ( )( )

Page 13: 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

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

− −

= − + + − + +

+ + −

( )( ) ( )( )( )

( )( )

Page 14: 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

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,

Page 15: 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

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

− −−

=+

+ −−

+ ++

Page 16: 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

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.

Page 17: 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

( )

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+ = + + + + +( )

Page 18: 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

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

+

+=

++

+