Question

Polinomul f = x^3+x^2+mx+m
Determinati valorile reale ale lui m stiind ca |x1|=|x2|=|x3|

Answer 1

$\displaystyle f(x)=x^2(x+1)+m(x+1)=(x^2+m)(x+1). \\ \\ O~radacina~este~x_1=-1,~deci~|x_1|=|x_2|=|x_3|=1. \\ \\ Celelalte~radacini~sunt~solutiile~ecuatiei~x^2+m=0 \Leftrightarrow x^2=-m. \\ \\ Deci~|x|^2=|m|. ~Insa~ |x|^2=1,~deci~|m|=1,~si~cum~m \in \mathbb{R},~rezulta~\\ \\m \in \{-1,1\}. \\ \\ \bullet Daca~m=-1,~atunci~x^2-1=0,~deci~\{x_2,x_3\}=\{-1,1\}.\\ \\ \bullet Daca~m=1,~atunci~x^2+1=0,~deci~\{x_2,x_3\}= \{-i,i\}. \\ \\ Deci~atat~m=-1,~cat~si~m=1~convin.$