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FactorTerms

Status: Stable

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

Description

FactorTerms[poly]
    pulls out any overall numerical factor in poly.
FactorTerms[poly, x]
    pulls out any overall factor in poly that does not depend on x.
FactorTerms[poly, {x1, x2, ...}]
    pulls out any overall factor in poly that does not depend on any of the xi, then progressively factors with respect to smaller subsets {x1, ..., x_{k-1}}.
FactorTerms[poly, x] extracts the content of poly with respect to x.
FactorTerms automatically threads over lists, equations, inequalities and logic functions.

Examples

All examples below are verified against the current Mathilda build.

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

In[2]:= FactorTerms[3 + 3a + 6 a x + 6 x + 12 a x^2 + 12 x^2, x]
Out[2]= 3 (1 + a) (1 + 2 x + 4 x^2)

In[3]:= FactorTerms[12 a^4 + 9 x^2 + 66 b^2]
Out[3]= 3 (4 a^4 + 22 b^2 + 3 x^2)

In[4]:= FactorTerms[7 x + (14 y + 21)/z]
Out[4]= (7 (3 + 2 y + x z))/z

In[5]:= FactorTerms[{5 x^2 - 15, 7 x^4 - 77, 8 x^8 - 24}]
Out[5]= {5 (-3 + x^2), 7 (-11 + x^4), 8 (-3 + x^8)}

In[6]:= FactorTerms[1 < 77 x^3 - 21 x + 35 < 2]
Out[6]= 1 < 7 (5 - 3 x + 11 x^3) < 2

In[7]:= f = 2 x^2 y z + 2 x^2 y + 4 x^2 z + 4 x^2 + 4 y^2 z^2 + 4 z y^2
Out[7]= 4 x^2 + 2 x^2 y + 4 x^2 z + 2 x^2 y z + 4 y^2 z + 4 y^2 z^2

In[8]:= FactorTerms[f, {x, y}]
Out[8]= 2 (1 + z) (2 x^2 + x^2 y + 2 y^2 z)

Implementation notes

Algorithm. builtin_factorterms (in src/poly/facpoly_factorterms.inc, compiled into facpoly.c) pulls out the content of a polynomial without factoring the polynomial part. It threads over List/equation/inequality/logic heads (ft_is_threading_head), then calls the shared engine ft_compute_list and multiplies the resulting factor list back into a single Times.

ft_compute_list first Together-normalises and splits into numerator/denominator. It collects and sorts the numerator's variables, then: (1) extracts the numerical content via ft_content_wrt_set (content with respect to all variables over an empty ground ring, i.e. the integer GCD of coefficients) and divides it out with ft_divide_out (exact polynomial division, falling back to symbolic Times[poly, content^{-1}]); (2) for a 2-arg call FactorTerms[poly, {x_1,…,x_k}], peels the content with respect to progressively smaller variable subsets, where ft_content_wrt_set recursively computes the multivariate poly_gcd_internal of the coefficients of each monomial in the chosen variables over the shrinking ground ring; (3) appends the final residue (re-multiplied by 1/den to round-trip rational inputs).

Data structures. Expr* polynomials throughout; poly_gcd_internal (multivariate polynomial GCD) is the workhorse for symbolic content; variable lists are Expr** sorted with compare_expr_ptrs.

  • Protected.
  • Auto-threads over List, Equal, Unequal, Less, LessEqual,

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

In[1]:= FactorTerms[2 x^2 + 4 x + 2]
Out[1]= 2 (1 + 2 x + x^2)

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

Notes

FactorTerms[poly] pulls out the overall numerical content (the integer GCD of the coefficients) without touching the polynomial structure — unlike Factor, it never splits 1 + 2 x + x^2 into (1 + x)^2. With a second argument FactorTerms[poly, x] extracts the content with respect to x, i.e. the factor that does not depend on x: here 3 y is the part free of x, leaving x^2 + 2 x y.