WISKUNDE
GRAAD 11
NOG OEFENINGE
  
Algebraiese breuke.
  
MATHEMATICS
GRADE 11
MORE EXERCISES
  
Algebraic fractions.
  
  
                Vereenvoudig die volgende breuke :
  
                 x2 ─ 3x
       1.  ─────────
              x2 ─ x ─ 6                                                   [ 1 ]
  
  
                6x2 ─ 21x
       3.  ─────────
             2x2 ─ 3x ─ 14                                              [ 3 ]
  
  
  
               2x2 + 5x ─ 12
       5.  ───────────
             2x2 ─ 11x + 12                                          [ 5 ]
  
  
  
               ax ─ bx + ay ─ by             x2 ─ 2xy + y2
       7.  ─────────────  X  ───────────
                       x2 ─ y2                     a2 ─ 2ab + b2
  
                                                                               [ 7 ]
  
  
                Simplify the following fractions :
  
                 x2 ─ 3x
       2.  ─────────
              x2 ─ x ─ 6                                               [ 2 ]
  
  
  
               x2 + 5x + 6
       4.  ─────────
               x2 + x ─ 6                                              [ 4 ]
  
  
  
               x2 ─ 2x ─ 24
       6.  ──────────
                  2x2 + 8x                                             [ 6 ]
  
  
  
               x2 ─ 2xy ─ 3y2           x2 + 2xy ─ 3y2
       8.  ───────────  X  ───────────
                  x2 ─ 9y2                 2x2 ─ 3xy + y2
  
                                                                             [ 8 ]
  
  
  
               ax ─ ay + bx ─ by             ax ─ ay ─ bx + by             ax + bx + ay + by
       9.  ─────────────   X   ─────────────   X   ─────────────
               ax + ay ─ bx ─ by                     a2 ─ b2                     ax ─ ay ─ bx + by                                            [ 9 ]
  
  
  
                    a2 + 8a + 15                  ab ─ 3a ─ 2b + 6              ab + 3a ─ 5b ─ 15
      10.  ─────────────   X   ─────────────   X   ───────────────
                ab + 3a ─ 2b ─ 6                  a2 ─ 2a ─ 15                 ab ─ 3a + 5b ─ 15                                     [ 10 ]
  
  
                 a2 ─ a ─ 6              a2 ─ 2a ─ 3                a2 + 2a + 1
      11.  ─────────   X   ──────────   X   ──────────
                 a2 + 3a + 2             a2 + 5a + 6                 a2 + 4a + 4                                                                 [ 11 ]
  
  
                 p2 ─ p ─ 2                 p2 ─ 3p + 2               p2 ─ 4p + 4
      12.  ──────────   X   ──────────   ÷   ──────────
                p2 + 5p + 6                   p2 ─ 1                     p2 + 4p + 3                                                              [ 12 ]
  
  
                  2x2 ─ 3x + 1             2x2 ─ 7x + 3               x2 ─ 3x + 2
      13.  ──────────   ÷   ──────────   ÷   ──────────
                2x2 ─ 3x ─ 2               x2 + x ─ 6               2x2 ─ 5x ─ 3                                                              [ 13 ]
  
  
                 6a2 ─ 7ab ─ 3b2             2a2 + ab ─ 6b2               9a2 ─ 3ab ─ 2b2
      14.  ─────────────   ÷   ────────────   ÷   ─────────────
                2a2 ─ 3ab ─ 2b2             3a2 ─ 8ab + 4b2              2a2 ─ 3ab ─ 2b2                                         [ 14 ]
  
  
                  2x2 + xy ─ y2                        x2 ─ y2                      3x2 ─ xy ─ 2y2
      15.  ─────────────   ÷   ────────────   X   ────────────
                4x2 + 4xy ─ 3y2                2x2 + 3xy ─ 2y2              6x2 + xy ─ 2y2                                            [ 15 ]
  
  
                3p2 + 8pq + 4q2                 4p2 ─ 5pq ─ 6q2              4p2 + 11pq + 6q2
      16.  ─────────────   X   ──────────────   ÷   ─────────────
                2p2 ─ 7pq + 6q2              10p2 ─ 11pq + 3q2             4p2 ─ 8pq + 3q2                                    [ 16 ]
  
  
  
                  a ─ b           a + b
      17.  ──────  +  ──────
                2a ─ b          a ─ 2b                                   [ 17 ]
  
  
  
                  a ─ b           a + b
      19.  ──────  ─  ──────
                2a ─ b          a ─ 2b                                   [ 19 ]
  
  
  
                 a + b             2b                 2a ─ b
      21.  ──────  +  ──────  ─  ──────
                a ─  b             a + b            a2 ─ b2
                                                                                   [ 21 ]
  
  
  
  
  
               3p + 2q          p ─ 3q
      18.  ──────  +  ──────
                 p + 3q          2p + q                           [ 18 ]
  
  
  
               3p + 2q          p ─ 3q
      20.  ──────  ─  ──────
                 p + 3q          2p + q                           [ 20 ]
  
  
  
               p + 2q            2p + q             3p ─ 2q
      22.  ──────  ─  ───────  +  ───────
               p ─ 2q           p2 ─ 4q2            p + 2q
  
                                                                           [ 22 ]
  
  
  
  
                      x + 3                    x ─ 2                     x ─ 3
      23.  ─────────   +   ───────   ─   ─────────────
                x2 + 3x + 2               x2 ─ 1                x2 + x ─ 2                                                                             [ 23 ]
  
  
  
                      x + 4                    x ─ 3                     x + 1
      24.  ─────────   +   ───────   ─   ─────────────
                x2 + x ─ 2               x2 ─ 4                x2 ─ 3x + 2                                                                             [ 24 ]
  
  
  
                      y ─ 1                     y ─ 6                    2y ─ 3
      25.  ─────────   ─   ─────────   +   ───────
                y2 + 5y + 6              y2 + y ─ 6                y2 ─ 4                                                                               [ 25 ]
  
  
  
                      2a + 1                         a ─ 5                       3a + 2
       26.  ──────────   +   ───────────   ─   ───────
                6a2 + 5a ─ 6              6a2 ─ 13a + 6              4a2 ─ 9                                                                     [ 26 ]
  
  
  
                         a ─ 3b                            a + b                           2a ─ 3b
       27.  ────────────   ─   ───────────   +   ───────────
                2a2 ─ 3ab ─ 2b2           a2 ─ 3ab + 2b2              2a2 ─ ab ─ b2                                                     [ 27 ]
  
  
  
                        6                             1
      28.  ─────────  +  ─────────
               x2 ─ 2x ─ 8           x2 + 5x + 6              [ 28 ]
  
  
  
                  1                  1                3a
      30.  ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2              [ 30 ]
  
  
  
                  2y                  1                1
      32.  ─────  ─  ─────  ─  ──────
               4 ─ y2           y ─ 2          2 + y              [ 32 ]
  
  
  
                     1               3                1
      34.  ──────  +  ───  +  ───────
               a(a ─ b)         ab          b(b ─ a)            [ 34 ]
  
  
  
  
                    x ─ 3                       x ─ 1
      29.  ─────────  ─  ──────────
               x2 ─ 3x ─ 4            x2 ─ x ─ 2           [ 29 ]
  
  
  
                  4                  3                7x
      31.  ─────  +  ─────  +  ──────
               1 + x           1 ─ x          x2 ─ 1              [ 31 ]
  
  
  
                  5                  2             6(1 ─ p)
      33.  ─────  ─  ─────  +  ──────
               3 + p           3 ─ p           p2 ─ 9            [ 33 ]
  
  
  
                  1                  3                1
      35.  ──────  +  ───  +  ───────
               p(p ─ 2)        2p          2(2 ─ p)            [ 35 ]
  
  
  
  
                        6                             1
      28.  ─────────  +  ─────────
               x2 ─ 2x ─ 8           x2 ─ 2x ─ 8              [ 28 ]
  
  
  
                  1                  1                3a
      30.  ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2              [ 30 ]
  
  
  
                  2y                  1                1
      32.  ─────  ─  ─────  ─  ──────
               4 ─ y2           y ─ 2          2 + y              [ 32 ]
  
  
  
                     1               3                1
      34.  ──────  +  ───  +  ───────
               a(a ─ b)         ab          b(b ─ a)            [ 34 ]
  
  
  
  
                    x ─ 3                       x ─ 1
      29.  ─────────  ─  ──────────
               x2 ─ 2x ─ 8            x2 ─ 2x ─ 8           [ 29 ]
  
  
  
                  4                  3                7x
      31.  ─────  +  ─────  +  ──────
               1 + x           1 ─ x          x2 ─ 1              [ 31 ]
  
  
  
                  5                  2             6(1 ─ p)
      33.  ─────  ─  ─────  +  ──────
               3 + p           3 ─ p           p2 ─ 9            [ 33 ]
  
  
  
                  1                  3                1
      35.  ──────  +  ───  +  ───────
               p(p ─ 2)        2p          2(2 ─ p)            [ 35 ]
  
  
  
  
                       5                            1                     x ─ 2
      36.  ─────────   +   ───────   ─   ────────
                18x + 54                54 ─ 18x           3x2 ─ 27                                                                              [ 36 ]
  
  
  
                2x ─ 1             2x + 1             4x + 2
      37.  ──────   +   ──────   +   ──────
                x2 + x              x2 ─ x             x ─ x3                                                                                             [ 37 ]
  
  
  
                1 ─ 2a             1 + 2a            1 ─ 20a2
      38.  ──────   +   ──────   −   ───────
                1 + 2a             1 ─ 2a            4a2 ─ 1                                                                                           [ 38 ]
  
  
  
               8 ─ 3a2             5a + 7                    2
      39.  ──────   −   ────────   +   ─────
               1 ─ a3              a2 + a + 1            a ─ 1                                                                                          [ 39 ]
  
  
  
  
  
  
  
  
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