Mathemagic: How to find all roots of polynomial if one imaginary root is given.

1.      Function: SUPPOSE THE POLYNOMIAL IS :  g(x) = x³ − 2x² − 11x + 52

Zero: 3 + 2i

        
          Remember always imaginary zeros always comes in pair.

So 3-2i is also a zero of g(x).

Hence {x-(3+2i)} and {x-(3-2i)} are factors of g(x).

And also g(x) will be divisible by product of these two factors.

Product of {x-(3+2i)} and {x-(3-2i)} is {(x-3)-2i}*{(x-3)+2i} = (x-3)²-(2i)²

=x²-6x+9-4i² = x²-6x+9+4 = x²-6x+13

Now by synthetic division,

(x²-6x+13 )   )  x³ − 2x² − 11x + 52 (   x+4

                                      

                                            X³-6x²  +13x

                                       ------------------------

                               +4x²-24x+52

                                 4x²-24x+52

                                       ------------------------            
                                                     0

                                                       ------------------------

Hence (x+4) is a factor of g(x).

=> x=-4 is also a zero of g(x).

So all the zeros of g(x) are: 3+2i, 3-2i, -4