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I have two expressions:

expr = (a - R)^2 (a + 2 R)/(2 R^3);
factor = 1 - a^2/R^2;

I would like to write expr in terms of factor -- is there a function in Mathematica to help with this?

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up vote 4 down vote accepted

Rewrite your definitions as equations (i.e. using ==), then use Solve to solve for expr specifying that the R and a variables should be eliminated:

Clear[expr, factor]

Solve[
  {expr == (a - R)^2 (a + 2 R)/(2 R^3), factor == 1 - a^2/R^2},
  expr,
  {R, a}
]

Mathematica graphics

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Using Eliminate

Eliminate[{expr == e, factor == f}, {a, R}]
3 f^2 + f^3 == (8 - 4 e) e
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3  
Alternatively: GroebnerBasis[{expr == (a - R)^2 (a + 2 R)/(2 R^3), factor == 1 - a^2/R^2}, {expr, factor}, {a, R}]. – J. M. Feb 25 at 15:04

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