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f1 = a x^2 + b x + c + x^3

(* a x^2+b x+c+x^3 *)

f1 /. x -> (x - a/3) // Expand

(* (2 a^3)/27-(a^2 x)/3-(a b)/3+b x+c+x^3 *)

I've tried this:

f1 /. x -> (x - a/3) // Expand // Collect[#, x] &

(* (2 a^3)/27+x (b-a^2/3)-(a b)/3+c+x^3 *)

f1 /. x -> (x - a/3) // Expand // Collect[#, x] & // PolynomialForm

(* (2 a^3)/27-(b a)/3+x^3+c+(b-a^2/3) x *)

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    $\begingroup$ You can extract the coefficients of $x$ as a list using f1 /. x -> (x - a/3) // CoefficientList[#, x] &. $\endgroup$ Commented Nov 19, 2013 at 17:36

2 Answers 2

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Try this formatting

OrderedForm[x_] := HoldForm[+##] & @@ (x^#1[[1]] #2 & @@@ CoefficientRules[#, x]) &;

f1 /. x -> (x - a/3) // OrderedForm[x]
x^3+(-(a^2/3)+b) x+((2 a^3)/27-(a b)/3+c)

See also my answer here.

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The OP almost had it with attempting to use PolynomialForm[]. All that was missing is the use of an appropriate setting:

PolynomialForm[a x^2 + b x + c + x^3 /. x -> (x - a/3), 
               TraditionalOrder -> True]
   (x - a/3)^3 + a (x - a/3)^2 + b (x - a/3) + c

As is usual with *Form[] functions, this is only suitable for display/pretty printing.

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