Question

sa se calculeze (x₁+1)²⁰¹⁶ +(x₂+1)²⁰¹⁶ stiind ca x este radacina a ecuatiei x²+x+1 =0
multumesc

Answer 1

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Answer 2

[tex]\it x^2+x+1=0\ \ \ \ \ (1) \\ \\ x^2+x+1=0 \Rightarrow (x-1)(x^2+x+1)=0 \Rightarrow x^3-1=0 \Rightarrow \\ \\ \Rightarrow x^3=1 \ \ \ \ \ (2)[/tex]

[tex]\it Pentru \ x_1,\ x_2\ r\breve{a}d\breve{a}cini\ pentru \ (1) \stackrel{(2)}{\Longrightarrow} x^3_1=x^3 _2=1\ \ \ \ \ (3) \\ \\ (1) \Rightarrow x_1 ^2 +x_1+1=0 \Rightarrow x_1+1=-x_1 ^2 \ \ \ \ \ (4) \\ \\ (x_1+1)^{2016}\ \stackrel{(4)}{=}\ (-x_1 ^2)^{2016} = x_1^{2\cdot 2016} =x_1^{4023} =x_1^{4032} = \\ \\ = x_1^{3\cdot1344} = [(x_1)^3]^{1344}\ \stackrel{(3)}{=}\ 1^{1344}=1[/tex]

[tex]\it Analog\ (x_2+1)^{2016} =1 \\ \\ \\ Deci,\ \ (x_1 +1)^{2016}+(x_2+1)^{2016} = 1+1=2[/tex]