Japan | A Nice Algebra Problem | Math Olympiad

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Synthetic division seems easier: d(x)=x^2+x+1=0; rem((1+x+x^2+x^3+x^4)/d(x))=(x+1) , rem(x^3/d(x))=1; x^49 = x*(x^3)^16) =x*1^16=x;
Now have x^49*(1+x+x^2+x^3+x^4) = x*(x+1) = -1