SOLUTION: Find an expression for cos(7A) and hence Solve: {{{P(x) = 64x^6 + 64x^5 - 48x^4 - 48x^3 + 8x^2 + 8x + 1}}} A warning in advance, this Q is really hard, I spent an hour on it and s

Consider complex number  

    z = cos(a) + i*sin(a).


Raise in degree 7.  You will get

     = .


Left side, due to de Moivre formula, is  cos(7a) + i*sin(7a).  So, you have

    cos(7a) + i*sin(7a) = .


In the right side, decompose using the Newton's binomial formula.

On the way, use identities   = -1;   = -i;   = 1 and so on.


Next, separate real and imaginary parts.  From real part, you will get the expression for cos(7a) via cos(a) and sin(a);

in this expression, all sin(a) will be in even degrees.

Exclude sin(a) in even degrees in this expression, using the formula  = .


In this way, you will get the expression

    cos(7a) = f(cos(a)),      (1)

where f(x) is some polynomial of x.


This polynomial is exactly what you are looking for.


Now, to find the root of this polynomial, notice that cos(7a) = 0 if  a = .


Therefore,  x=   is the root of the polynomial (1).


By the way, its other roots are  ,  ,  ,  ,  ,  and  ,  due to the same reason (!)