Goodies Polynomials
A polynomial is good if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius 0.99 on the complex plane that contains all the roots.Is there is any good polynomial for a sufficient large degree?
BONUS: Prove the conjecture whatever your answer maybe
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1 solution
The polynomial is of the form ( x − r 1 ) . . . ( x − r n ) . All the roots have magnitude strictly less than 1. The constant term is the product of the roots. It has magnitude less than 1. So it must be zero. So a root must be zero, say r 1 = 0 . So the polynomial is of the form x ( x − r 2 ) . . . ( x − r n ) . Repeat this argument again for the coefficient of x . None of the r i can be zero since the roots are distinct, so we have a contradiction.
Maybe I made a mistake ...