Is that much info enough?
What is the necessary condition for the value of b such that all the complex roots of x 4 − 8 x 3 + 2 4 x 2 + b x + c = 0 are positive reals?
The answer is -32.
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1 solution
that is a very good solution...+1
Why must c = 1 6 ?
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2 must be a root with multiplicity 4, so that the function must be f ( x ) = ( x − 2 ) 4
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What I mean to say that I could have c = 1 0 and the equation have at least one non-real number .
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@Pi Han Goh – I trust that what Ayush means is that all the complex roots are positive reals. I will add that for clarity.
By the way, Comrade, I know that I owe you (at least) one answer. I'm a little busy (and excited) as I get ready to travel to Cambridge, MA, and then to Cambridge, UK.
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@Otto Bretscher – I don't understand you. All complex roots are positive reals, the converse is not true.
Take your time, there's no hurry to answer my question.
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@Pi Han Goh – Oh I get it now... he writes "necessary" because he does not specify the constant c . So, "necessary" is necessary after all... I'm not paying attention.
Since f ′ ′ ( x ) = 1 2 ( x − 2 ) 2 ≥ 0 , the graph of f ( x ) is strictly convex, so that f ( x ) has at most two simple real roots or at most one real multiple root. Thus we must have f ( x ) = ( x − 2 ) 4 = x 4 − 8 x 3 + 2 4 x 2 − 3 2 x + 1 6 , so that b = − 3 2 .