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Cubics

Status: Experimental

present and registered, but lightly documented and not yet covered by dedicated tests.

Description

Cubics is an option for Solve that controls whether cubic equations are solved via explicit radical formulas (Cubics -> True) or returned as held Root[] objects (default Cubics -> False).

Examples (2)

Every input below was run against the current Mathilda build and its output recorded.

Applications (2)

In[1]:= Solve[x^3 + x + 1 == 0, x]
Out[1]= {{x -> Root[1 + #1 + #1^3 &, 1]}, {x -> Root[1 + #1 + #1^3 &, 2]}, {x -> Root[1 + #1 + #1^3 &, 3]}}

In[2]:= Solve[x^3 + x + 1 == 0, x, Cubics -> True]
Out[2]= {{x -> -1/3 ((1/2 (27 + 3 Sqrt[93]))^(1/3) - 3/(1/2 (27 + 3 Sqrt[93]))^(1/3))}, {x -> -1/3 ((-1)^(2/3) (1/2 (27 + 3 Sqrt[93]))^(1/3) + (3 (-1)^(1/3))/(1/2 (27 + 3 Sqrt[93]))^(1/3))}, {x -> -1/3 (-(3 (-1)^(2/3))/(1/2 (27 + 3 Sqrt[93]))^(1/3) - (-1)^(1/3) (1/2 (27 + 3 Sqrt[93]))^(1/3))}}

Implementation notes

Cubics is not a function ā€” it is an option symbol for Solve/Roots (and forwarded by the eigenvalue solver). It has no builtin handler; it is interned in sym_names.c and registered only with a docstring. Option parsing in solve.c (is_*_option / the option setter) reads Cubics -> True|False into opts->poly.cubics_radical, which controls whether degree-3 factors are solved by radicals (Cardano) or returned as held Root[] objects. The actual radical-vs-Root decision lives in the polynomial solver (src/poly/solvepoly.c); the default is Cubics -> False.

Attributes: none registered.

References

See also: Solve

Notes & additional examples

Notes

Cubics is a Solve option, not a function. With the default Cubics -> False, irreducible cubics are returned as held Root[] objects; setting Cubics -> True forces the explicit Cardano radical formulas, as the contrast above shows.