are the seven roots of the equation . Their cosines are the x-coordinates of the seven coloured dots in the attached picture, which are equally spaced around the unit circle. Notice that the blue dots have the same x-coordinates as the red dots.

There is a formula for in terms of , namely . So the equation can be written as The seven roots of this equation are (each of which is a repeated root), the seventh root being .

The next step is to factorise the left side of that equation as

The c+1 factor obviously corresponds to (the green dot in the picture) and the three roots of the squared factor correspond to the x-coordinates of the red (or blue) dots. Thus the equation has roots Since , we can put and conclude that the equation has roots The sum of the roots of the equation is 4, so that is the answer to the problem.