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Suppose , , and are nonzero real numbers, and . What are the possible value(s) for ?

Number Theory

Easy

Excursion in Mathematics

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Try to think elementarily . That means from the given relation \( a+ b +c =0 \) , what can be the possible signs of the real numbers , \( a, b , \ and \ c \) accordingly. Then proceed .

There are cases to consider: Case : 2 of , , and are positive and the other is negative. With out loss of generality, we can assume that and are positive and (as the relation is symmetric) is negative. In this case, we have that , Think about the another similar case .

Case : 2 of , , and are positive and the other is negative. Here also without loss of generality, we can assume that and are negative and is positive. In this case, we have that

In both cases, we get that the given expression equals .

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