SOLUTION: What are the sum of the coefficient of Q(x), when 37x^73-73x^37+36 is divided by x-1.

     Let Q(x) be a quotient of division the polynomial P(x) =   by binomial (x-1).
     Find the sum of coefficients of the polynomial Q(x).

1.  Notice that  x=1 is the root of the polynomial P(x) = .
    It is easy to check directly:  P(1) = 0.

    Hence, the polynomial P(x) is divisible by the binomial (x-1) without a remainder, and the quotient  is a polynomial with integer coefficients.

    Again, the quotient Q(x) =  is a polynomial with integer coefficients, and the problem asks to find the sum of the coefficients of this polynomial.


2.  For any polynomial F(x), the sum of its coefficients is equal to the value of the polynomial at x=1:

    the sum of coefficients of F(x) is equal to F(1).


3.  Based on it, the first idea is to calculate Q(1) by substituting directly x=1 into the quotient .
    But it doesn't work, since the denominator becomes 0 (zero) at x=1.


4.  Although this crude idea doesn't work, there is another way to calculate Q(1) as the limit of Q(x) at x-->1.

    Notice that Q(x) =  ==  = , 

    therefore  Q(1) = limit of  at x-->1  = (derivative of P(x) at x=1)  =  (P'(x) at x=1) = P'(1).


    P'(x) is easy to express. It is :  P'(x) = 

    and therefore P'(1) = 37*73 - 73*37 = 0.


5.  Thus we have this chain of equalities:

    sum of coefficients Q(x) = Q(1) = (lim Q(x) at x-->1)  = P'(1) = 0.


Answer.  The sum of coefficients of the quotient  is 0 (zero).