Transform an exact polynomial

Normalize and round-trip exact coefficients

Leading zeros disappear, while rational coefficients stay exact. Record contains everything needed to reconstruct the same canonical polynomial.

Runnable RiXRun cell

.Plugin.Load("algebra"); P := .algebra.Polynomial([0, 0, 1, -6, 11, -6]); Copy := .algebra.Polynomial(P.Record()); .Table( ["property", "exact value"], [ ["coefficients", P.Coefficients()], ["P(2)", P(2)], ["round trip equal", .algebra.Equal(P, Copy)] ] );

Verify a factor and show synthetic division

The transformation retains exact quotient/remainder objects and factor metadata. Its Grid is the same portable layout family as intrinsic .Algebra.SyntheticDivision, so document and terminal renderers can display it.

Runnable RiXRun cell

.Plugin.Load("algebra"); P := .p`x^3 - 6x^2 + 11x - 6`; Factor := .p`x - 2`; division := .algebra.SyntheticDivide(P, 2); quotient := division.Quotient().Coefficients(); remainder := division.Remainder().Coefficients(); isFactor := P.IsFactor(Factor); division.Grid();

Inspect exact factor evidence

The evidence distinguishes discovered rational factors from the residual and verifies exact reconstruction rather than relying on approximate roots.

Runnable RiXRun cell

.Plugin.Load("algebra"); P := .p`(x-1)^2*(x+2)^3`; evidence := .algebra.FactorEvidence(P); .Table( ["property", "exact value"], [ ["rational roots", evidence[:rationalRoots]], ["factor records", evidence[:factors]], ["residual", evidence[:residual].Coefficients()], ["verified", evidence[:verified]] ] );

Use .algebra.SquareFreeDecomposition(P) when multiplicity groups matter, .algebra.Gcd(P, Q) or .algebra.Lcm(P, Q) for canonical combinations, and .algebra.Resultant(P, Q) for an exact shared-factor test.

Keep centered and factored forms explicit

Presentation values preserve a useful form without creating a second notion of Polynomial equality. Expanding either value reconstructs from its displayed fields and verifies the exact result against the retained canonical source.

Runnable RiXRun cell

.Plugin.Load("algebra"); P := .poly([2, -3, 5, -7]); centered := .algebra.CenteredExpansion(P, 2); factored := .algebra.Factorization(.p`6*(x-1)^2*(x+2)^3*(x^2+1)`); .Table( ["property", "exact value"], [ ["powers of (x-2), ascending", centered.Coefficients()], ["centered round trip", centered.Expand().Coefficients()], ["factorization unit", factored.Unit()], ["rational factors", factored.Factors()], ["monic residual", factored.Residual().Coefficients()], ["all factors rational", factored[:complete]], ["factorization round trip", factored.Polynomial().Coefficients()] ] );

Record() produces a portable form accepted by .algebra.Expand(record). Changing a coefficient, root, factor, multiplicity, unit, residual, basis, or completeness claim causes conversion to fail rather than silently returning the stored source.

Decompose a rational function exactly

The Algebra façade also exposes .ratfun Phase 2 transformations. Repeated rational poles become individual exact terms; factors not split over Q remain visible in one proper residual rather than being approximated.

Runnable RiXRun cell

.Plugin.Load("algebra"); R := .rf`(x^5+x^3+2*x+1)/((x-1)^2*(x^2+1))`; partial := .algebra.PartialFractions(R); divisor := .algebra.PoleZeroEvidence(R); .Table( ["property", "exact value"], [ ["coefficient domain", .algebra.CoefficientDomain(R)[:id]], ["polynomial part", partial.PolynomialPart().Coefficients()], ["linear-pole terms", partial.Terms()], ["proper residual", partial.Residual()], ["verified round trip", .algebra.Expand(partial.Record()) == R], ["pole evidence", divisor.Poles()] ] );