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Interesting Polynomial

Let a be sum and b be product all real roots of the equation \(x^{4}-x^{3}-1=0\). Prove \(b<\frac{-11}{10}\) and \(a>\frac{6}{11}\).

Note by Minh Triết Lâm
4 years ago

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Note that the polynomial can be written, thanks to our dear friend Gauss (among others), as \(k(x-r_1)(x-r_2)(x-r_3)(x-r_4)=k(x^4-x^3(r_1+r_2+r_3+r_4)+\cdots +(-1)^4 r_1r_2r_3r_4))\). Thus \(b\) is the coefficient of \(x^0\) in the polynomial (as \(k=1\)) and \(-a\) is the coefficient of \(x^3\).

A L - 4 years ago

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