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

Find all polynomials P(x)=anxn+an1xn1+a1x+a0P(x) = a_n x^n + a_{n-1} x^{n-1} \dots + a_1 x + a_0 such that an0a_n \neq 0, (an,an1,,a1,a0)(a_n, a_{n-1}, \dots, a_1, a_0) is a permutation of (0,1,2,,n)(0, 1, 2, \dots, n) and all zeroes of P(x)P(x) are in Q\mathbb{Q}.

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

#Algebra

Note by Michael Tong
6 years, 10 months ago

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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.

ww margera - 6 years, 8 months ago
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