Given the cubic trinomial equation $$ x^3+2x^2+2x+1=0 $$ , find the value of $$ x^{1994}+x^{1997}+x^{2000} $$

Question

Given the cubic trinomial equation  $$ x^3+2x^2+2x+1=0 $$ , find the value of  $$ x^{1994}+x^{1997}+x^{2000} $$

Uniteasy Admin · Accepted Answer

Factoring the cubic polynomial to solve the function.
$$ x^3+2x^2+2x+1=0 $$ 
$$ (x^3+x^2)+(x^2+2x+1)=0 $$
$$ x^2(x+1)+(x+1)^2=0 $$
$$ (x+1)(x^2+x+1)=0 $$
$$ (x+1)(x^2+x+\dfrac{1}{4}+\dfrac{3}{4} )=0 $$
$$ (x+1)\Big[ (x+\dfrac{1}{2})^2+\dfrac{3}{4}\Big] =0 $$
$$ 	herefore x=-1 $$
$$ x^{1994}+x^{1997}+x^{2000} $$
$$ =1 $$

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