MODERATE EXERCISES

(1) Find the constants A and B so that the given equation is true on its domain.

 3x2 + 7x + 1   A     B        C
 ------------ = - + ----- - -------
  x(x + 1)2     x   x + 1   (x + 1)2

Since  3x2 + 7x + 1   A     B        C
       ------------ = - + ----- - -------
        x(x + 1)2     x   x + 1   (x + 1)2 
We can find a common denominator on the left to get:

    3x2 + 7x + 1    A(x + 1)2 + Bx(x + 1) - Cx
    ------------ = ---------------------------
     x(x + 1)2              x(x + 1)2


Therefore we see that 3x2 + 7x + 1 = Ax2 + 2Ax + A + Bx2 + Bx - Cx

                                 = (A + B)x2 + (2A + B - C)x + A

   Therefore
   A = 1
   2A + B - C = 7  ==>  2(1) + B - C = 7
   A + B = 3  ==>  1 + B = 3  ==>  B = 2

   And 2 + 2 - C = 7  ==>  C = -3

So we get:

       3x2 + 7x + 1   1     2       -3
       ------------ = - + ----- - -------
         x(x + 1)2    x   x + 1   (x + 1)2

       3x2 + 7x + 1   1     2        3
       ------------ = - + ----  + -------
          x(x+1)2     x   x + 1   (x + 1)2