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Omni-MATH / Does there exist positive reals a₀,a₁,…,a₁₉, such that the polynomial P(x)=x²⁰+a₁₉x¹⁹+…+a₁x+a₀ does not have any real roots, yet all polynomials formed from swapping any two coeffi…
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problem
Does there exist positive reals , such that the polynomial does not have any real roots, yet all polynomials formed from swapping any two coefficients has at least one real root?
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Does there exist positive reals a₀,a₁,…,a₁₉, such that the polynomial P(x)=x²⁰+a₁₉x¹⁹+…+a₁x+a₀ does not have any real roots, yet all polynomials formed from swapping any two coefficients a_(i),a_(j) has at least one real root?
Original LaTeX notation
Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?Discussion
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