What is a good technique for solving polynomials?Say for example: 6 x 3 − 17 x 2 − 4 x + 3 = 0

Question

What is a good technique for solving polynomials?Say for example:  6      x          3        −  17      x          2        −  4  x  +  3  =  0

tiltat9h · Accepted Answer

First, to minimize the lead coefficient, we reverse the polynomial (which reciprocates  the roots) and then we scale it to make the leading coefficient 1, i.e. we apply the AC-method, which yields                    f        (        x        )                                    =                            3                          x          3                −        4                          x          2                −        17                x        +        6                            ⇒                          9                f        (        x        )                                    =                            (        3        x                  )          3                −        4                (        3        x                  )          2                −        51                (        3        x        )        +        54                                          =                                                z          3                −                  4                                  z          2                −        51                z        +                  54                ,                                   z        =        3        x            Note     z  =      1     is a root. By Vieta, other roots have product       −    54          /              1        ,   sum       4      −        1    =  3  ,   so are     9  ,    −  6.  Hence     z    =    1  ,    9  ,    −  6     so     x    =    z      /    3    =    1      /    3  ,    3  ,    −  2  ,   which reciprocated yields     3  ,    1      /    3  ,    −  1      /    2.