Question

x^3-5*x+1. Calc suma inverselor soluțiilor.

Answer 1

   
[tex]\displaystyle\\ \text{Se da polinomul: }~~~~~f = x^3-5x+1\\ \text{Solutiile ecuatiei:}~~~~~x^3-5x+1=0~~~~~\text{sunt: }~~ x_1,~~x_2,~~x_3 \\\\ \text{Inversele solutiilor sunt: }~~ \frac{1}{x_1},~~\frac{1}{x_2},~~\frac{1}{x_3}\\\\ \text{Suma inverselor solutiilor este: }\\\\ \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3} = ~~\text{(Aducem fractiile la acelasi numitor)}\\\\ =\frac{x_2 x_3}{x_1x_2 x_3}+\frac{x_1x_3}{x_1x_2 x_3}+\frac{x_1x_2}{x_1x_2 x_3}=\frac{x_2 x_3+ x_1x_3+x_1x_2}{x_1x_2 x_3}\\\\ [/tex]


[tex]\displaystyle\\ \text{Folosim Relatiile lui Viete pentru polinomul de gradul 3: }\\\\ \text{Daca avem polinomul:} ~~f=ax^3+bx^2+cx+d\\\\ \text{Relatiile lui Viete sunt:}\\\\ x_1+x_2 +x_3= \frac{-b}{a}\\\\ x_1x_2+x_2 x_3+ x_1x_3=\frac{c}{a}\\\\ x_1x_2 x_3= \frac{-d}{a}\\\\ \text{Polinomul nostru este: }~~f=x^3-5x+1~~\text{in care:}\\ a=1\\ b=0\\ c=-5\\ d=1\\\\ \frac{x_2 x_3+ x_1x_3+x_1x_2}{x_1x_2x_3}=\frac{\dfrac{c}{a}}{\dfrac{-d}{a}}=\frac{\dfrac{-5}{1}}{\dfrac{-1}{1}}= \frac{-5}{-1} =\boxed{\bf 5} [/tex]

Answer 2

O alta metoda mult mai simpla .