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What is wrong with this.

I want to show this:Let a_1,...,a_n be positive real numbers. Prove that the polynomial

P(x) = x^n-a_1 x^{n-1}-...-a_n has a unique positive zero.

Q(x) = x^n+a_1 x^{n-1} + ...+ a_n has n complex nonzero zeroes. For each of them, we have that

0 = |x^n+a_1 x^{n-1} + ...+ a_n | \geq ||x|^n-a_1 |x|^{n-1} - ...- a_n |

which implies that |x| is a zero y^n-a_1 y^{n-1} - ...- a_n. But that implies that there could be more than one unless it is somehow true that all of the zeros of Q(x) lie on a circle in the complex plane!

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# What is wrong with this

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