Behavior around Zeros

 Example 11:   Algebraically   

f(x) = (x + 2) (x + 1) (x - 6)^2

Find the zeros of this function, then describe the behavior of the function around the zeros.
    Zeros
        x + 2 = 0       x = -2

        x + 1 = 0        x = -1
        x - 6 = 0       x = 6

   Number line
          Using test points around the zeros, determine the behavior around these points.
            -3            -1.5               0                7           Test Pts
        <------------(-2)---------------(-1)---------------(6)-------------->

  Plug in test points in f(x)
        x = -3            (-3 + 2) (-3 + 1) (-3 - 6)^2             =   (-)(-)(+)    =     +    positive
        x = -1.5         (-1.5 + 2) (-1.5 + 1) (-1.5 - 6)^2      =   (+)(-)(+)    =     -    negative
        x = 0             (0 + 2) (0 + 1) (0 - 6)^2                 =   (+)(+)(+)    =     +    positive
        x = 7             (7 + 2) (7 + 1) (7 - 6)^2                 =    (+)(+)(+)   =     +    positive

          -3            -1.5               0                7           Test Pts
        <++++++++++(-2)---------------(-1)+++++++++++++(6)+++++++++++++>

 Describe the behavior around the zeros
  (-∞, -2) ∪ (-1, 6) ∪ (6, ∞)      All points are above the x-axis   (Positive)
  There is a double root at x = 6
  (-2, -1)         All points are below the x-axis    (Negative)