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Let A and B be real numbers such that \(\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}\). What is A+B?

 Mar 19, 2020
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Write the two term on the left-hand side as one fraction, with denominator (x - 5):

     A / (x - 5) + B(x + 1)  =  A / (x - 5) + B(x + 1)(x - 5) / (x - 5)  =  [A + B(x + 1)(x - 5)] / (x - 5)

 

Multiplying out, and simplifying,  the numerator:

     [A + Bx2 - 4Bx - 5B] / (x - 5)     =     [ Bx2 - 4Bx +(A - 5B) ] / (x - 5)

 

Setting the numerator on the left side with the numerator on the right side:

     Bx2 - 4Bx +(A - 5B)    =   -3x2 + 12x + 22

 

Setting the two x-squared terms equal to each other:  Bx2  =  -3x2     --->     B  =  -3

 

Setting the two x terms equal to each other:  -4Bx  =  12x     --->     -4B  =  12     --->     B  =  -3

 

Setting the two constant terms equal to each other:  A - 5B  =  22     --->     A - 5(-3)  =  22

                                                                        --->     A + 15  =  22     --->     A  =  7

 Mar 19, 2020

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