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Collect

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Collect[expr, x]
    expands expr and gathers terms with the same power of x, returning
    a sum of the form Sum[c_k x^k] with each c_k free of x.
Collect[expr, {x1, x2, ...}]
    collects with respect to each xi in turn (nested grouping).
Collect[expr, x, f]
    applies f to each coefficient before re-assembling the sum, useful
    for f = Simplify or f = Factor.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Collect[a x + b y + c x + d y, x]
Out[1]= (a + c) x + b y + d y

In[2]:= Collect[a x^4 + b x^4 + 2 a^2 x - 3 b x + x - 7, x]
Out[2]= -7 + (1 + 2 a^2 - 3 b) x + (a + b) x^4

In[3]:= Collect[a Sqrt[x] + Sqrt[x] + x^(2/3) - c x + 3x - 2b x^(2/3) + 5, x]
Out[3]= 5 + (1 + a) Sqrt[x] + (1 - 2 b) x^(2/3) + (3 - c) x

Implementation notes

Algorithm. builtin_collect (in src/poly/poly.c) groups an expression by powers of one or more keys, delegating to the recursive worker collect_internal. For each key it expands the expression with respect to that key (expr_expand_patt), except when the key is itself a Plus — there expansion would distribute the subterm and is skipped so e.g. Collect[a(c+s)+b(c+s), c+s] stays grouped. Each summand is decomposed into base–power form; terms are bucketed by the exponent at which the key appears (single-base keys group by the rational/symbolic exponent ratio, multi-factor monomial keys by the integer multiplicity from get_k). The collected coefficients are summed and, if a third head argument h is given, wrapped with h. It threads over List, equations and inequalities (skipping operator slots in Inequality), and recurses across multiple keys.

Data structures. Base–power lists (as in Coefficient) for term decomposition; results are rebuilt with internal_times/Plus and re-evaluated through eval_and_free.

  • Protected.
  • Automatically threads over lists, equations, inequalities, and logic functions.
  • Effectively writes expr as a polynomial in x or a fractional power of x.
  • Collect[expr, var, h] applies h to the expression that forms the coefficient of each term obtained.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Collect[a x^2 + b x^2 + c x + x, x]
Out[1]= (1 + c) x + (a + b) x^2

In[1]:= Collect[(1 + x + y)^3, x]
Out[1]= 1 + x^3 + 3 y + 3 y^2 + y^3 + x (3 + 6 y + 3 y^2) + x^2 (3 + 3 y)

In[1]:= Collect[Expand[(1 + x)^4 (1 + y)^2], x, Factor]
Out[1]= (1 + y)^2 + 4 x (1 + y)^2 + 6 x^2 (1 + y)^2 + 4 x^3 (1 + y)^2 + x^4 (1 + y)^2

In[1]:= Collect[(x + y + z)^2, {x, y}]
Out[1]= x^2 + y^2 + 2 y z + z^2 + x (2 y + 2 z)

Notes

Collect[expr, x] expands expr and gathers terms by power of x, leaving each coefficient free of x. A third argument applies a function (e.g. Factor or Simplify) to each coefficient before reassembly; a list of variables groups by each in turn.