# 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, 8 months 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 $$x^3$$.

- 4 years, 8 months ago