Find all polynomials P(x)=anxn+an−1xn−1⋯+a1x+a0P(x) = a_n x^n + a_{n-1} x^{n-1} \dots + a_1 x + a_0P(x)=anxn+an−1xn−1⋯+a1x+a0 such that an≠0a_n \neq 0an=0, (an,an−1,…,a1,a0)(a_n, a_{n-1}, \dots, a_1, a_0)(an,an−1,…,a1,a0) is a permutation of (0,1,2,…,n)(0, 1, 2, \dots, n)(0,1,2,…,n) and all zeroes of P(x)P(x)P(x) are in Q\mathbb{Q}Q.
I don't know the answer to this, but it is a problem we can work on together.
Note by Michael Tong 6 years, 10 months ago
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2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
We can at least prove that the 0 coefficient must be at the constant place. Here is how:
Suppose that 0 is at a non constant position:
From the fact that P(0) >= 0 and P'(0) >= 0, we can deduce that all the roots are strictly negative. So the elementary symmetric polynomial of degree d in the roots is negative if d is odd and positive if d is even. In other words, it can never be zero, as is required for 0 to be at a non-constant position.
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Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
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or_italics_
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or__bold__
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[example link](https://brilliant.org)
> This is a quote
\(
...\)
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to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
We can at least prove that the 0 coefficient must be at the constant place. Here is how:
Suppose that 0 is at a non constant position:
From the fact that P(0) >= 0 and P'(0) >= 0, we can deduce that all the roots are strictly negative. So the elementary symmetric polynomial of degree d in the roots is negative if d is odd and positive if d is even. In other words, it can never be zero, as is required for 0 to be at a non-constant position.