SOLUTION: find all roots of the polynomial x3 - x2 +16x -16 possible answers: 1, 4, -4 -1, 4, -4 -1, 4i, -4i 1, 4i, -4i what does the i come from? please explain equation and solut

find all roots of the polynomial x^3 - x^2 +16x -16



possible answers:

1, 4, -4

-1, 4, -4

-1, 4i, -4i

1, 4i, -4i



what does the i come from? please explain equation and solution.Thank you



To solve for the roots of the polynomial(sometimes called zeros of the polynomial)

Let us find possible factors of x^3 - x^2 +16x -16, p/q

                                 |               |

                                 q               p

 Where p = 16 (possible factor) = 1, -1, 4, -4, 16, -16

       q = 1  (possible factor) = 1, -1



The possible factors of x^3 - x^2 +16x -16 are

 p/q = 1, -1, 4, -4, 16, -16



Now we can use synthetic division to check which are the true factor.

   factor        Coefficients

1|            1  -1  16  -16

                 1   0   16

            _______________ 

             1   0  16   0 --> since no reminder, 1 is a factor of f(x)



Then lets find other factors aside from x - 1

             the quotient 1  0 16 means x^2 + 16 



Solve for x to find other factors of  x^3 - x^2 +16x -16:



   x^2 + 16 = 0

        

           Remember 

             x = ±4i



       

Therefore the roots of the polynomial are

1, 4i, -4i