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Let be the set S of all real numbers $\alpha$ such that the function

\[\frac{x^2 + 5x - \alpha}{x^2 - 7x - 44}\]

can be expressed as a quotient of two linear functions. What is the sum of the elements of S?

 May 30, 2021
 #1
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Factoring the denominator:

$$\frac{x^2+5x-\alpha}{(x+11)(x-4)}$$

Thus $(x+11)\mathrm{ or }(x-4)|(x^2+5x-\alpha)$.

Hence either $x^2+5x-\alpha=(x+11)(x+t)$ or $x^2+5x-\alpha = (x+t)(x-4)$.

The first case, sum of roots is $-5=-11-t\implies 5=11+t\implies t=-6$ and hence $(x+11)(x-6)=x^2+5x-66$ thus $\alpha = 66$. 

The second case, sum of roots is $-5=4-t\implies t=9$ and hence $\alpha=36$.


The answer is then $66+36=102$.

 May 30, 2021

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