caffeineinmyveins
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- May 22, 2018
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I recently came across a problem that was marked as a hard one (and the book does a good job at picking hard problems).
Find a cubic polynomial whose roots are cos{2pi/7}, cos{4pi/7}, and cos{6pi/7}.
The solution given in the solutions manual painstakingly constructs the polynomial 8x^3+4x^2-4x-1, but without any other restrictions on the problem, why can't you just make something like (x-cos{2pi/7})(x-cos{4pi/7})(x-cos{6pi/7})? I don't understand. Do you think the authors were missing some extra conditions, like the condition that the coefficients had to be integers? Or is there something wrong with the trivial solution?
Find a cubic polynomial whose roots are cos{2pi/7}, cos{4pi/7}, and cos{6pi/7}.
The solution given in the solutions manual painstakingly constructs the polynomial 8x^3+4x^2-4x-1, but without any other restrictions on the problem, why can't you just make something like (x-cos{2pi/7})(x-cos{4pi/7})(x-cos{6pi/7})? I don't understand. Do you think the authors were missing some extra conditions, like the condition that the coefficients had to be integers? Or is there something wrong with the trivial solution?