Hello, Sensei!

Greetings.

I gather you’re here about that quartic.

Indeed. $x^4 -2x^3 - x^2 +2x+1$. You seemed to think that I wouldn’t spot it was a perfect square?

Oh! Did I? Gosh, that doesn’t sound like the sort of thing I would say. Are you sure you’re not confusing me… yeah, ok, it’s a fair cop.

One can, of course, complete the square on a quartic.

Can one?

*But of course. One can always reduce it to a perfect square plus at most a linear term. You see, $(ax^2 + bx + c)^2 = a^2 x^4 + 2ab x^3 + (2ac+b^2) x^2 + 2bc x + c^2$.

I’m very glad you come with subtitles, sensei, I’d never have followed that out loud. It looks legit – you have freedom to pick $a$, $b$ and $c$, so you can alwyas make the first three terms match whatever you’ve got.

Legit.

OK, OK, put the eyebrow down. So with our quartic, $x^4 -2x^3 - x^2 +2x+1$, we’ve got $a = 1$ immediately, then $b = -1$ and $c=-1$ – so it’s $(x^2 - x - 1)^2$ plus something that might be linear.

So far so good.

And then our remaining terms are exactly what we expect, so it’s a perfect square. And I’ll return the bow right now before you van… oh, they’ve already left.

I’m not sure how useful this is in general – unless the $x$ term does vanish, you’re not going to be able to use it to solve the quartic directly.

All the same, it’s a faintly useful quick-and-dirty trick and who knows? Maybe one day it’ll save you from a Ninja attack.