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Find constants A and B such that for all x such that x not equal to 2 and 4, (2x + 3)/(x^2 - 6x + 8) = A/(x - 2) + B/(x - 4).  Give your answer as the ordered pair (A,B).

 Jun 30, 2020
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(2x + 3) / (x2 - 6x + 8)  =  A / (x - 2) + B / (x- 4)

 

Since  x2 - 6x + 8  can be factored as  (x - 2)(x - 4),

multiply each term by  (x - 2)(x - 4)  to get:     2x + 3  =  A(x - 4) + B(x - 2)

                                                                        2x + 3  =  Ax - 4A + Bx - 2B

 

Set the x-terms equal to each other:  2x  =  Ax + Bx     --->    2x  =  (A + B)x     --->   A + B  =  2

 

Set the constants equal to each other:  3  =  -4A - 2B   --->   4A + 2 B  =  -3

 

Solve this set of two equations:     A +   B  =    2     to get your answer.

                                                     4A + 2B  =  -3  

 Jun 30, 2020

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