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# Find all of the polynomials with the following qualities...

Find all polynomials $$P(x) = a_n x^n + a_{n-1} x^{n-1} \dots + a_1 x + a_0$$ such that $$a_n \neq 0$$, $$(a_n, a_{n-1}, \dots, a_1, a_0)$$ is a permutation of $$(0, 1, 2, \dots, n)$$ and all zeroes of $$P(x)$$ are in $$\mathbb{Q}$$.

I don't know the answer to this, but it is a problem we can work on together.

Note by Michael Tong
3 years, 5 months ago

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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.

- 3 years, 3 months ago