Question

Polinoame

Sa se demonstreze ca f nu are toate radacinile reale

f= 3x^4 - 2x^3 + x^2 +ax -1

Answer 1

$f = 3x^4 - 2x^3 + x^2 +ax -1\\ \\ S_1 = x_1+x_2+x_3+x_4=\dfrac{-b}{a} \\ S_2 = x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4 = \dfrac{c}{a}\\ \\ x_1^2+x_2^2+x_3^2+x_4^2 = S_1^2-2S_2 = \\ \\ =\Big(-\dfrac{-2}{3}\Big)^2-2\cdot \dfrac{1}{3} = \dfrac{4}{9}-\dfrac{2}{3}= -\dfrac{2}{9} <0 \\ \\ \Rightarrow x_1^2+x_2^2+x_3^2+x_4^2 < 0 \Rightarrow \exists~x_k\in \mathbb{C}\backslash \mathbb{R},~~k = \overline{1,4}$