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find the common roots of these equations. z 3 +2z 2 +2z+1=0 z 1985 +z 100 +1=0 (a) w, w 2 (b) 1, w, w 2 (c) -1, w, w 2 (d) -w, -w 2
the answer is option A.W+W^2+W^3=0 and W^3=1 are the formulas required to solve this problem.substitute W in equation 1,we get W^3+2W^2+2W+1=0which implies 1+2(W^2+W)+1=0W^2+W=-W^3=-1HENCE THIS BECOMES 2+2(-1)=0.hence W is root of given equation.similarly try with W^2.now substitute W in equation 2.the equation becomes W^1985+W^100+1=0NOW DIVIDE 1985 BY 3.we get 2 as remainder.divide 100 by 3.we get 1 as remainder.now the equation becomes W^3*661(W^2)+W^3*33(W)+1=0AS WE KNOW THAT W^3=1now the equation becomes W^2+W+1=0Hence proved.similarly try with W^2
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