A strange equation V

Algebra Level 5

The seven real roots of the equation x7+28x6+308x5+1680x4+4704x3+6272x2+3136x1282+256=0x^7+28x^6+308x^5+1680x^4+4704x^3+6272x^2+3136x-128\sqrt{2}+256 = 0 are, in ascending order: x1=abcos(cπd)x_{1}=-a-b\cos\left(\dfrac{c\pi}{d}\right) x2=abcos(eπd)x_{2}=-a-b\cos\left(\dfrac{e\pi}{d}\right) x3=abcos(fπd)x_{3}=-a-b\cos\left(\dfrac{f\pi}{d}\right) x4=abcos(gπd)x_{4}=-a-b\cos\left(\dfrac{g\pi}{d}\right) x5=a+bcos(hπd)x_{5}=-a+b\cos\left(\dfrac{h\pi}{d}\right) x6=a+jjx_{6}=-a+j\sqrt{j} x7=a+bcos(kπd)x_{7}=-a+b\cos\left(\dfrac{k\pi}{d}\right) All the variables are positive integer numbers, and the angles are in radians between 00 and π2\frac{\pi}{2}. Find a+b+c+d+e+f+g+h+j+ka+b+c+d+e+f+g+h+j+k.

You may also try Part VI.

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