Question 1196201
.



            It is  VERY  nice problem,  admitting  BEATIFUL  solution.

            See below.



<pre>
If x^2 = 1-x, then

   x^8 = (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4.


The idea is to replace in this expression, and in all expressions that follow, x^2 by (1-x) 
everywhere, where it is possible, and as many times as possible, until you get the desired 
expression of degree 1 (one).


                        See how it works


    x^8 = 1 - 4x + [6(1-x)] - [4x*(1-x)] + [(1-x)*(1-x)] = I continue =

        = 1 - 4x + 6 - 6x - 4x + 4x^2 + 1 - 2x + x^2 = 8 - 14x + 5x^2 = I replace x^2 by (1-x) again = 

        = 8 - 14x + 5*(1-x) = 8 - 16x + 5 - 5x = 13 - 21x.    

 
    So, p = 13, q = 21  and  p + q = 13 + 21 = 34.      <U>ANSWER</U>
</pre>

Solved.


This method is called &nbsp;" the lowering of a degree " &nbsp;method.



//////////////



It can be solved by different methods, &nbsp;but this one is a &nbsp;" true delight ".