This computation leads to the so-called "depressed cubic" which is a cubic polynomial in ax + b with no quadratic term. And one can continue in this same way to arrive at "depressed" quartics, quintics, etc. They're all polynomials in ax + b, with no subleading term.
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And since all these "depressed" polynomials are polynomials in ax + b, we know that when the coefficient sequence a, b, c, ... is geometric, they must take the form (ax + b)^n = 0 -- which means that their coefficients must all vanish in this case. Like ac - b^2, ad - bc, etc.
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Indeed, one easily finds, for example, that a^2(ax^3 + 3bx^2 + 3cx + d) = (ax + b)^3 + 3(ac - b^2)(ax + b) + a(ad - bc) - 2b(ac - b^2). And this "depressed" cubic in ax + b has for its coefficients polynomials in ac - b^2 and ad - bc -- both zero in the geometric case.
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I thought you might enjoy this string of ideas, which one can easily make into a "general theory of depression." It goes farther than this, too, as I imagine will not surprise you. Really quite lovely stuff, which puts "completing the square" into its proper algebraic context.
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A final remark: in this context, the cubic formula takes on a peculiarly elegant and memorable form: if ax^3 + 3bx^2 + 3cx + d = 0, then ax + b = cbrt[(ac - b^2)(ar + b)] + cbrt[(ac - b^2)(as + b)], where r, s are the roots of (ac - b^2)x^2 + (ad - bc)x + (bd - c^2) = 0.
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Replying to @MathPrinceps
@threadreaderapp God wants polynomials written a certain way, and I suspect he'd also like you to unroll the thread.2 replies 0 retweets 1 like -
Replying to @80k_0k8 @threadreaderapp
I'm embarrassed never to have learned how to do this. I suppose I'd better amend that situation.
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Replying to @MathPrinceps @threadreaderapp
0h_0k Retweeted Thread Reader App
0h_0k added,
Thread Reader App @threadreaderappReplying to @80k_0k8Hi there is your unroll: Thread by@MathPrinceps: "@jamestanton God wants us to write our polynomials like this: 1ax^0 1ax^1 + 1bx^0 1ax^2 + 2bx^1 + 1cx^0 1ax^3 + 3bx^2 + […]" https://threadreaderapp.com/thread/1102050069228937216.html … Have a good day.
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Replying to @80k_0k8 @threadreaderapp
Thank you very much. I am slowly becoming less ignorant.
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Replying to @MathPrinceps @threadreaderapp
My Twitter suggestion is to separate out those who are always attacking people from those who are willing to have a reasonable discussion. I've found that math people on Twitter are generally helpful, or at least have some nice perspective to offer.
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I've formed a similar impression. Although even among the mathematicians using Twitter I have noticed some who seem exceptionally sensitive to any slight, however unintentional, and who are prone to an unforgiving stance, no matter the apologies proffered. Nerve-wracking for me.
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