Which linear expression is a factor of the polynomial function f(x) = x3 + 2x2 − x − 2? x − 2 x + 2 x + 3 x − 3

Please, use " ^ " to denote exponentiation:

f(x) = x^3 + 2x^2 − x − 2

Also, please separate your answer choices with a comma or semicolon, or put each on a new line by itself:

x − 2
x + 2
x + 3
x − 3

The easiest way to choose the correct linear expression is to use synthetic division.  If you do this, and find that you have no remainder, then you know that a particular divisor in synth. div. is a root, which leads to a factor.

Look at x-2.  This is a possible factor.  The associated possible root is 2. Use 2 as the divisor in synth. div.:
   
2 /  1   2   -1   -2
           2    8    14
   
      1   4    7     12           Since there is a non-zero remainder, we conclude                                              that 2 is not a root and (x-2) is not a factor.

You could also use long division and carry on until you find a possible factor for which the remainder is zero.