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Expand

Status: Stable

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

Description

Expand[expr] expands out products and powers in expr.
Expand[expr, patt] leaves unexpanded any parts of expr that are free of the pattern patt.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Expand[(x+3)(x+2)]
Out[1]= 6 + 5 x + x^2

In[2]:= Expand[(x+y)^2 (x-y)^2]
Out[2]= x^4 - 2 x^2 y^2 + y^4

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

Implementation notes

Algorithm. builtin_expand (in src/expand.c) calls expr_expand_patt(e, patt), a structural recursion that distributes products over sums. The dispatch by head:

  • Plus — expand each summand and rebuild via eval_and_free.
  • Times — expand each factor, then multiply them pairwise with multiply_all, a divide-and-conquer fold whose leaf operation multiply_two handles the four Plus×Plus / Plus×atom / atom×Plus / atom×atom cases, producing every cross term (a_i · b_j) and re-summing them.
  • Power[base, k] with k a positive integer < 100 — expand the base, then power_expand raises it by repeated squaring (binary exponentiation), each multiply going through the distributing multiply_two. (Note: there is no explicit binomial-coefficient formula; the binomial expansion of (a+b)^k falls out of the iterated distribution.) Negative or non-integer exponents are left untouched.
  • List / equations / inequalities / And / Or / Not — threads into each argument (passing operator-symbol slots of Inequality through verbatim).

A two-argument Expand[expr, patt] only expands subexpressions that contain patt (gated by expr_contains_patt, which uses the pattern matcher); everything else is copied unchanged. Expand does not descend into the arguments of arbitrary function heads.

Data structures. Plain Expr* trees; transient Expr** argument buffers per node. All arithmetic rebuilding goes back through the evaluator (eval_and_free) so Plus/Times canonicalisation (Flat/Orderless, like-term collection) applies after each step.

Complexity / limits. Worst case is the full cross-product blow-up of distribution: expanding (x_1+...+x_m)^k produces O(m^k)-sized output, and Power is capped at exponent < 100 to bound it.

  • Protected.
  • Works only on positive integer powers and distributes products over sums.
  • Threads over equations, inequalities, and lists.
  • Implements an efficient binary-splitting algorithm for distributing products and repeated squaring for powers.
  • Expand[expr, patt] leaves unexpanded any parts of expr that are free of the pattern patt.

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]:= Expand[(x + 1)^3]
Out[1]= 1 + 3 x + 3 x^2 + x^3

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

In[1]:= Expand[(a + b)(c + d)]
Out[1]= a c + b c + a d + b d

In[1]:= Expand[(x + 2)^2 (x - 1)]
Out[1]= -4 + 3 x^2 + x^3

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

In[1]:= Expand[(1 + x)^10]
Out[1]= 1 + 10 x + 45 x^2 + 120 x^3 + 210 x^4 + 252 x^5 + 210 x^6 + 120 x^7 + 45 x^8 + 10 x^9 + x^10

Notes

Expand applies the distributive law to products and integer powers, producing a flat sum of monomials in canonical order (ascending total degree in the leading variable). Like terms are combined automatically, so (x + 2)^2 (x - 1) collapses the x^1 coefficient to zero and it drops out of the result. Expand only multiplies out — it does not factor or cancel — and a second argument Expand[expr, patt] leaves alone any parts free of the pattern patt.