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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?

 May 15, 2021
 #1
avatar+605 
+1

Note that by synthetic division $-3 x^2 + 12 x + 22 = (-3 x - 3)(x - 5) + 7 = -3(x+1)(x-5)+7$. Divide both sides by $x-5$.

 May 15, 2021
 #2
avatar+26397 
+1

Let A and B be real numbers such that
Ax5+B(x+1)=3x2+12x+22x5.
What is A+B?

 

Ax5+B(x+1)=3x2+12x+22x5|(x5)A+B(x+1)(x5)=3x2+12x+221. x=1:A+0=3(1)2+12(1)+22A=312+22A=72. x=2:7+B(1)(7)=3(2)2+12(2)+227+7B=1224+227B=1477B=21|:7B=3A+B=73A+B=4

 

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 May 15, 2021

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