What are the zeros of the function? f(x)=x3−2x2−3x −3 , 0, and 2 −2 , 0, and 3 −3 , 0, and 1 −1 , 0, and 3

Seeing that the last (the constant) term is -3, we could make the reasonable assumption that one of the zeros of this function may be among {-3, -1, 1, 3}.  Choosing any one of these possible zeros, we can do synthetic division to determine whethere or not the chosen number actually is a zero of this function.

     
-1  /  1   -2   -3    -3
              -1    3     0
     
        1    -3    0     -3        Since there is a non-zero remainder, we know
                                        that -1 is NOT a zero.

Is -3 a zero?  Let's try it:

     
-3  /  1   -2   -3    -3
              -3   15   -39
     
        1   -5    13   -42         No, -3 is not a zero, because the remainder is 
                                          not zero.

At this point I ask you to double check to ensure that you have copied this problem down correctly.  I graphed this function on my TI-83 calculator and have found that there is only one real root, and that root is not an integer.