But what's most interesting about all this is that the same ideas apply all the way along the sequence of God's preferred polynomials. In particular, the next step is to compute a(ax^3 + 3bx^2 + 3cx + d) - (ax + b)(ax^2 + 2bx + c), which, it turns out, is equally lovely.
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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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Laurens- Take a look at the result, it's much easier for me to read.https://threadreaderapp.com/thread/1102050069228937216.html …
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I have done so, and I quite agree that it is much easier to read -- not just for you, but objectively, for any reader. Again, my thanks for the help.
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