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Linear term on the denominator
14

Partial Fractions Linear Term

Jul 20, 2015

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Page 1: Partial Fractions Linear Term

Linear term on the denominator

Page 2: Partial Fractions Linear Term

Expressions with ax + b in the denominator decompose to terms of the form:

A

ax b

Page 3: Partial Fractions Linear Term

2

8 42

3 18

x

x x

To express

as partial fractions.

Page 4: Partial Fractions Linear Term

8 42

( 6)( 3)

x

x x

First factorise the denominator:

Page 5: Partial Fractions Linear Term

6 3

A B

x x

Express as individual fractions with the factors on the denominator:

Page 6: Partial Fractions Linear Term

( 3) ( 6)

( 6)( 3)

A x B x

x x

Express with a common denominator:

Page 7: Partial Fractions Linear Term

There are two methods to proceed from here.

Page 8: Partial Fractions Linear Term

Comparing coefficients

Page 9: Partial Fractions Linear Term

2

8 42 ( 3) ( 6)

3 18 ( 6)( 3)

x A x B x

x x x x

( ) 3 6

( 6)( 3)

A B x A B

x x

Page 10: Partial Fractions Linear Term

8 42 ( ) 3 6x A B x A BComparing the coefficients:

A + B = 8-3A + 6B = -42

Solving gives A = 10 and B = -2

Looking at the numerators:

Page 11: Partial Fractions Linear Term

2

8 42 10 2

3 18 ( 6) ( 3)

x

x x x x

Hence:

Page 12: Partial Fractions Linear Term

Substitution

Page 13: Partial Fractions Linear Term

8 42 ( 3) ( 6)x A x B x

Any value of x can be substituted, but in this case x = 3 and then x = -6, for obvious reasons, are best.

x = 3

24 – 42 = 9BB = -2

x = -6

-48 – 42 = -9AA = 10

Page 14: Partial Fractions Linear Term

2

8 42 10 2

3 18 ( 6) ( 3)

x

x x x x

Hence: