Editing Laguerre's method (section)

===Fallback===
Even if the "drastic set of assumptions" does not work well for some particular polynomial {{math|''p''(''x'')}}, then {{math|''p''(''x'')}} can be transformed into a related polynomial {{mvar|r}} for which the assumptions are viable; e.g. by first shifting the origin towards a suitable complex number {{mvar|w}}, giving a second polynomial {{math|''q''(''x'') {{=}} ''p''(''x'' − ''w'')}}, that give distinct roots clearly distinct magnitudes, if necessary (which it will be if some roots are complex conjugates).

After that, getting a third polynomial {{mvar|r}} from {{math|''q''(''x'')}} by repeatedly applying the root squaring transformation from [[Graeffe's method]], enough times to make the smaller roots significantly smaller than the largest root (and so, clustered comparatively nearer to zero). The approximate root from Graeffe's method, can then be used to start the new iteration for Laguerre's method on {{mvar|r}}. An approximate root for {{math|''p''(''x'')}} may then be obtained straightforwardly from that for {{mvar|r}}.

If we make the even more extreme assumption that the terms in <math>G </math> corresponding to the roots <math>x_2,\ x_3,\ \ldots,\ x_n </math> are negligibly small compared to the root <math>x_1,</math> this leads to [[Newton's method]].