Show that the sum of the third roots of 1 is zero? Thank you!
1 Answer
See explanation...
Explanation:
Suppose
#(x-alpha)(x-beta)(x-gamma)=x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma#
Note that the third roots of
#x^3+0x^2+0x-1#
So equating the coefficients of
Bonus
If you would like to explicitly find all the third roots of
#0 = x^3-1#
#color(white)(0) = (x-1)(x^2+x+1)#
#color(white)(0) = (x-1)((x+1/2)^2+3/4)#
#color(white)(0) = (x-1)((x+1/2)^2-(sqrt(3)/2i)^2)#
#color(white)(0) = (x-1)((x+1/2)-sqrt(3)/2i)((x+1/2)+sqrt(3)/2i)#
#color(white)(0) = (x-1)(x+1/2-sqrt(3)/2i)(x+1/2+sqrt(3)/2i)#
So the three roots are:
#1# ,#" "-1/2+sqrt(3)/2i" "# and#" "-1/2-sqrt(3)/2i#