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Find all A and B such that \(\frac{4x}{x^2-8x+15} = \frac{A}{x-3} + \frac{B}{x-5}\) for all x besides 3 and 5. Express your answer as an ordered pair in the form (A, B).

 

Not quite sure how to solve this. Thanks in advance.

Guest Mar 30, 2018
 #1
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    4x                                  A                   B

__________     =          ______   +       _____

x^2 - 8x + 15                  x  - 3               x - 5

 

This is known as a partial-fraction decomposition

 

Note that the denominator on the left factors as  ( x - 3)  (x  - 5)

Multiply both sides by this and we have

 

4x   =  A(x - 5)   +  B ( x - 3)     simplify

 

4x  = Ax - 5A + Bx - 3B

 

4x + 0 =  (A + B)x  + (- 5A -3B)

 

Equating coefficients, we have this system

 

A + B  = 4         ⇒  B  = 4 - A      (1)

-5A  - 3B  =  0   ⇒  5A  + 3B  =  0     (2)

 

Sub (1)  into (2)   and we have that

 

5A + 3(4 - A)  = 0

5A + 12 - 3A  = 0

2A  =  -12

A  = -6

 

And  B  =  4 - A   =   4 - (-6)   =  10

 

Verify  that

 

-6/ ( x - 3)  +  10/ (x  - 5)     =   the left hand side of the original equation

 

 

cool cool cool

CPhill  Mar 30, 2018

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