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Find the product CD of the integers C and D for which

C/(x - 3) + D/(x + 8) = (4x + 65)/(x^2 + 5x - 24)

for all real value of x except -8 and 3.

 Apr 13, 2022
 #1
avatar+122390 
+1

We can use the method of partial fractions

 

Note that   (x^2 + 5x - 24)  =   ( x +8) (x -3)         (1)

 

Multiply through by (1)    and we have that

 

4x + 65   =   C (x + 8)  + D(x -3)          simplify

 

4x + 65  = (C + D)x + (8C - 3D)          equate terms and we have these equations

 

C + D = 4         ⇒    D   =  4 - C     (2)

8C - 3D = 65       (3)

 

Sub (2)  into (3)  and we have

 

8C - 3(4 -C)  =65

8C - 12 + 3C  = 65

11C = 77

C = 7

and

D = 4 - 7   =  -3

 

C * D  = (7) (-3)   =  -21

 

 

cool cool cool

 Apr 13, 2022

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