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# Algebra

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Let S be the set of all real numbers a such that the function (x^2+5x+a)/(x^2-10x-24) can be expressed as the quotient of two linear functions. What is the sum of the elements of S?

May 15, 2022

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May 15, 2022
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$$\dfrac{x^2 + 5x + a}{x^2 - 10x - 24} = \dfrac{x^2 + 5x + a}{(x - 12)(x + 2)}$$

If the fraction is equal to the quotient of two linear functions, then either (x - 12) or (x + 2) divides x^2 + 5x + a.

By factor theorem,

$$12^2 + 5\cdot 12 + a = 0\text{ or }(-2)^2 + 5(-2) + a = 0\\ a = -204 \text{ or }a = 6$$

Therefore $$S = \{-204, 6\}$$. Sum of elements of S is -198.

Remarks: The statement of factor theorem is as follows.

Let $$p(x)$$ be a polynomial such that $$(x - a)$$ divides $$p(x)$$. Then $$p(a) = 0$$.

May 16, 2022
edited by MaxWong  May 16, 2022