5.5. Solved Problems

1. Perform the indicated operations and simplify:

      2         3       4x - 2       1
   ——————— + ——————— - ———————— = ———————
    x - 1     x + 1     x2 - 1     x - 1
   

2. Simplify

      1       1
    ————— - —————
    x - 1   x + 1     x + 1 - x + 1      2     1
   ——————————————— = ——————————————— = ———— = ———
      1       1       x + 1 + x - 1     2x     x
    ————— + —————
    x - 1   x + 1
   

3. Simplify:

   (x + δ)2 - x2    x2 + 2xδ + δ2 - x2 
   ————————————— = ————————————————— = 2x + δ
         δ                  δ 
   

4. Simplify:

   (x + δ)3 - x3    x3 + 3x2δ + 3xδ2 + δ3 - x3 
   ————————————— = ————————————————————————— = 3x2 + 3xδ + δ2 
         δ                   δ 
   

5. Write as the sum of a polynomial and a proper rational expression:

    x3 + x2 + x + 1     x2(x + 1) + (x + 1)     x2 + 1
   ———————————————— = ————————————————————— = ————————
      x2 + 2x + 1         (x + 1)(x + 1)        x + 1
   
                   x - 1
     x + 1 ) x² + 0x + 1
             x² +  x
                  -x + 1
                  -x - 1
                       2
   
                 2
    = x - 1 + ———————
               x + 1
   

6. Simplify and write as the sum of a polynomial and a proper rational expression:

    3x2 + 7x + 2    (3x + 1)(x + 2)    3x + 1
   ————————————— = ———————————————— = ———————— 
     x2 + x - 2      (x - 1)(x + 2)     x - 1
   
                3
    x - 1 )3x + 1
           3x - 3
                4
   
             4
    = 3 + ——————— 
           x - 1
   

7. Simplify and write as the sum of a polynomial and a proper rational expression:

    4x2 + 12x + 9        (2x + 3)2        2x + 3
   ——————————————— = ———————————————— = ———————— 
    2x2 + 11x + 12    (2x + 3)(x + 4)     x + 4
   
                2
    x + 4 )2x + 3
           2x + 8
               -5
   
             5
    = 2 - ———————
           x + 4
   

8. Simplify:

    4x2 - 5x - 21        (x - 3)(4x + 7)         4x + 7 
   ——————————————— = ————————————————————— = ————————————— 
       x3 - 27        (x - 3)(x2 + 3x + 9)     x2 + 3x + 9 
   

9. Simplify:

    6x2 - x - 1       (3x + 1)(2x - 1)     3x + 1
   ——————————————— = ————————————————— = ——————————
    2x3 - 3x2 + x     x(x - 1)(2x - 1)     x(x - 1)
   

10. Write as the sum of a polynomial and a proper rational expression:

       x4 - 81       (x2 + 9)(x - 3)(x + 3)    (x2 + 9)(x - 3)
    ————————————— = ——————————————————————— = ————————————————
     x2 - x - 12         (x - 4)(x + 3)            (x - 4)
    
        x3 - 3x2 + 9x - 27
     = ———————————————————
              x - 4
    
                   x² +  x + 13
      x - 4 )x³ - 3x² + 9x - 27
             x³ - 4x²
                   x² + 9x
                   x² - 4x
                       13x - 27
                       13x - 52
                             25
    
                       25
     = x2 + x + 13 + ———————
                      x - 4 
    

11. Perform the indicated operations and simplify:

     2x2 + 15x - 6     x2 - x + 1 
    ——————————————— + ——————————— = 
        x3 + 125        x2 + 125 
    
        3x2 + 14x - 5        (3x - 1)(x + 5)
     = ——————————————— = —————————————————————— =
           x3 + 125       (x + 5)(x2 - 5x + 25)
    
          3x - 1
     = ——————————————
        x2 - 5x + 25
    

12. Perform the indicated operations and simplify:

     x2 + 7x + 1     7x + 2
    ————————————— - ———————— = 
        x3 + 1       x3 + 1
    
       x2 - 1
    = ———————  = 
       x3 + 1
    
         (x + 1)(x - 1)
    = ———————————————————— = 
       (x + 1)(x2 - x + 1)
    
          x - 1 
    = ————————————
       x2 - x + 1 
    

13. Perform the indicated operations and write the result as the sum of a polynomial and a proper rational expression:

     5x2 - 30x + 10     x2 + 2x - 20 
    ———————————————— + —————————————— =
       x2 - x - 20       x2 - x - 20 
    
        6x2 - 28x - 10
     = ———————————————— = 
          x2 - x - 20
    
       (x - 5)(6x + 2)
     = ————————————————— = 
       (x - 5)(x + 4)
    
        6x + 2
     = ———————— = 
         x + 4
    
                  6
     x + 4 )6x +  2
            6x + 24
                -22
    
              22
     = 6 - ———————
            x + 4
    

14. Perform the indicated operations and simplify:

      2x3 - 2x         5x2 - 5
    ———————————— + ————————————— = 
    2x2 + 7x + 5    2x2 + 7x + 5
    
        2x3 + 5x2 - 2x - 5     (2x + 5)(x2 - 1)
     = ———————————————————— = ————————————————— = 
           2x2 + 7x + 5        (2x + 5)(x + 1)
    
       (2x + 5)(x + 1)(x - 1)
     = —————————————————————— = x - 1
           (2x + 5)(x + 1)
    

15. Perform the indicated operations and write the result as the sum of a polynomial and a proper rational expression::

       x - 1         x(2x - 7)          2x - 7  
    ——————————— + ——————————————— - —————————————————— = 
      3(x + 2)    (x + 2)(3x - 1)    3(x + 2)(3x - 1)
    
       (x - 1)(3x - 1) + 3x(2x - 7) - (2x - 7)
     = ——————————————————————————————————————— = 
                 3(x + 2)(3x - 1)
    
        3x2 - 4x + 1 + 6x2 - 21x - 2x + 7
     = —————————————————————————————————— = 
                 3(x + 2)(3x - 1)
    
          9x2 - 27x + 8     (3x - 8)(3x - 1)
     = ————————————————— = ————————————————— = 
        3(x + 2)(3x - 1)    3(x + 2)(3x - 1)
    
        3x - 8
     = ———————— = 
        3x + 6
    
                  1
     3x + 6 )3x - 8
             3x + 6
                -14
    
              14
     = 1 - ————————
            3x + 6
    

16. Perform the indicated operations and write the result as the sum of a polynomial and a proper rational expression:

     2x2 - 8     3x2 - 17x - 28 
    ————————— × ——————————————— = 
     3x + 4       2x2 + 6x + 4
    
        2(x + 2)(x - 2)     (x - 7)(3x + 4) 
     = ————————————————— × ———————————————— = 
           3x + 4           2(x + 2)(x + 1)
    
        (x - 2)(x - 7)      x2 - 9x + 14 
     = ————————————————— = —————————————— = 
            x + 1              x + 1
    
                  x - 10
     x + 1 )x² - 9x + 14 
            x² +  x
               -10x + 14
               -10x - 10
                      24
    
                   24 
     = x - 10 + ———————
                 x + 1
    

17. Perform the indicated operations and write the result as the sum of a polynomial and a proper rational expression:

        x3 + 8  
      ————————— 
        3x - 2 
    —————————————— = 
      x2 - 2x + 4 
     ———————————— 
        12x - 8 
    
       (x + 2)(x2 - 2x + 4)     4(3x - 2) 
     = ———————————————————— × ———————————— = 
           3x - 2              x2 - 2x + 4   
    
     = 4x + 8 
    

18. Perform the indicated operations and simplify:

       1         4        2 - i + 8 + 4i      10 + 3i
    ——————— + ——————— = —————————————————— = —————————
     2 + i     2 - i     4 - 2i + 2i - i2        5
    

19. Perform the indicated operations and simplify:

     1 + i     4 + i      4 + i + 4i - i2     3 + 5i
    ——————— × ——————— = —————————————————— = ————————
     2 + i     2 - i     4 - 2i + 2i - i2       5
    

20. Perform the indicated operations and simplify:

      1 + i 
     ———————
      2 - i     (1 + i)(4 - i)     4 -  i + 4i - i2     5 + 3i
    ————————— = ——————————————— = —————————————————— = ———————  
      3 + i     (2 - i)(3 + i)     6 + 2i - 3i - i2     7 - i
     ———————
      4 - i 
    
      (5 + 3i)(7 + i)     35 + 5i + 21i + 3i2
    = ———————————————  = ————————————————————  
      (7 -  i)(7 + i)     49 + 7i -  7i -  i2
    
       32 + 26i
    = ——————————
          50 
    

21. Find the first four partial quotients for e and evaluate the continued fraction that gives the approximate value.

    r = e = 2.718282
    a0 = floor(r) = 2
    p0 = 2×1 + 0 = 2
    q0 = 2×0 + 1 = 1
    
    r = 1/(r - a0) = 1/(2.718281 - 2) = 1.392211
    a1 = floor(r) = 1
    p1 = 1×2 + 1 = 3
    q1 = 1×1 + 0 = 1
    
    r = 1/(r - a1) = 1/(1.392211 - 1) = 2.549647
    a2 = floor(r) = 2
    p2 = 2×3 + 2 = 8
    q2 = 2×1 + 1 = 4
    
    r = 1/(r - a2) = 1/(2.549647 - 2) = 1.819350
    a3 = floor(r) = 1
    
                 1                 1              3           3
    x = 2 + ———————————— = 2 + ————————— = 2 + ——————— = 2 + ——— = 2.75
                   1                 1          3 + 1         4 
            1 + ————————        1 + ———
                     1               3
                2 + ———
                     1