INTRODUCTORY EXERCISES

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

 (a)    9x + 21          A         B
     -------------- = ------- + -------
     (x + 5)(x - 3)    x + 5     x - 3


 (b)      3x2 + x          A      Bx + C
      --------------- = ------ + -------
      (x - 2)(x2 + 3)    x - 2    x2 + 3

 (a)    9x + 21      A(x - 3) + B(x + 5)
    -------------- = -------------------
     x2 + 2x - 15       x2 + 2x - 15

                      Ax - 3A + Bx + 5B
                   = -------------------
                        x2 + 2x - 15 


                      (A + B)x + (5B - 3A)
                   = ----------------------
                        x2 + 2x - 15

   Next, we solve for the unknowns in the two equations we obtain:

            A + B = 9
          5B - 3A = 21

   We find that A = 3 and B = 6.


 (b)   3x2 + x        A      Bx + C
    ------------- = ----- + -------
    (x -2)(x2 + 3)  x - 2    x2 + 3

                    A(x2 + 3) + (Bx + C)(x-2)  
                  = ------------------------- 
                         (x - 2)(x2 + 3)
 

                   Ax2 + 3A + Bx2 - 2Bx + Cx - 2C
                  = ------------------------------
                            (x - 2)(x2 + 3)


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

   Next, we solve for the unknowns in the three equations we obtain:

   A + B = 3 ===> B = 3 - A
   -2B + C = 1 ===> -2(3 - A) + C = 1 ===> -6 + 2A + C = 1 ===> C = 7 - 2A
   3A - 2C = 0 ===> 3A - 2(7 - 2A) =0 ===> 3A - 14 +4A = 0 ===> 7A = 14
   Therefore A = 2 ,B =3 - 2 =1 and C = 7 - 2(2) = 3