In particular, when the coefficients a, b, c, ... are in geometric progression (i.e., b = ar, c = ar^2, etc.) then so are these polynomials. Indeed, the nth polynomial is then just a(x + r)^n. So in this case. all the roots coincide: all n of them are equal to -r = -b/a.
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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