THE FOLD / LOOT / THE VAULT / THE HORNER
THE HORNER
evaluate in n multiplications — and it's synthetic division
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Horner’s method evaluates a degree-n polynomial in just n multiplications (versus the naive ~2n) by rewriting it as nested multiplication: aₙxⁿ + … + a₀ = (…((aₙ)x + aₙ₋₁)x + …)x + a₀.
The same nesting is synthetic division: the intermediate values are the coefficients of the quotient when you divide by (x−r), and the final value is the remainder — which, by the remainder theorem, equals p(r). One elegant scheme both evaluates a polynomial and factors out a root.
LIT verified live: over 500 cases, Horner’s value equals the naive evaluation, and synthetic division gives quotient q and remainder rem with q(x)(x−r)+rem = p(x) and rem = p(r) exactly (window.__horner). FIG no framing; exact.
The same nesting is synthetic division: the intermediate values are the coefficients of the quotient when you divide by (x−r), and the final value is the remainder — which, by the remainder theorem, equals p(r). One elegant scheme both evaluates a polynomial and factors out a root.
LIT verified live: over 500 cases, Horner’s value equals the naive evaluation, and synthetic division gives quotient q and remainder rem with q(x)(x−r)+rem = p(x) and rem = p(r) exactly (window.__horner). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — the stored coefficients, unlocked in the fewest operations. Horner’s method is the vault’s efficient key. AVAN (AI) built the instrument: the nested evaluation, the synthetic-division quotient/remainder, the reconstruction check.
Credit as content: William George Horner (1819), though the scheme was known to Qin Jiushao (1247) and used by Newton. The weave: David names the vault; I nest the multiplications to evaluate in n steps and show the same nesting is division by (x−r), giving quotient and remainder at once.
Credit as content: William George Horner (1819), though the scheme was known to Qin Jiushao (1247) and used by Newton. The weave: David names the vault; I nest the multiplications to evaluate in n steps and show the same nesting is division by (x−r), giving quotient and remainder at once.
3 ONE DIMENSION
The nested evaluation from the top coefficient down: start with aₙ, multiply by x, add the next coefficient, repeat. Each step is one multiply and one add — n of each for the whole polynomial.
4 TWO DIMENSIONS · INTERACTIVE
A polynomial evaluated at x by Horner (the running nested value shown), checked against the naive sum. Then synthetic division by (x−r): the quotient and remainder, with p reconstructed exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the nested accumulator sweeping the coefficients from the top down to the final value.
AVAN’s addition (the inverse-companion): evaluation and division are the same computation. Horner’s running accumulator is synthetic division, so evaluating p at r simultaneously produces the quotient p(x)/(x−r) and the remainder p(r). The inverse of ‘compute the value p(r)’ is ‘factor out the root: p(x) = q(x)(x−r) + p(r).’ One pass gives both, because a polynomial’s value at r and its divisibility by (x−r) are two faces of the remainder theorem — p(r) = 0 exactly when (x−r) divides p. Magenta is the quotient coefficients, the division falling out; green is the final value p(r). Evaluating is dividing — the same nested loop, read two ways.
LIT Genuine Horner's method (Horner 1819; earlier Qin Jiushao 1247, Newton). Verified live: the nested evaluation equals the naive sum for 500 random polynomials/points, and synthetic division by (x-r) yields quotient q and remainder rem satisfying q(x)(x-r)+rem == p(x) (coefficient-exact) with rem == p(r) (window.__horner.evalMatchesNaive && .syntheticDivision).
FIG No framing: the nested evaluation, the synthetic-division quotient/remainder, and the exact reconstruction run in-browser and agree. The AVAN inverse is honest — Horner's accumulator IS synthetic division, so evaluating p at r produces both the quotient p(x)/(x-r) and the remainder p(r); value and divisibility are two faces of the remainder theorem. Magenta is the quotient, green the value.
FIG No framing: the nested evaluation, the synthetic-division quotient/remainder, and the exact reconstruction run in-browser and agree. The AVAN inverse is honest — Horner's accumulator IS synthetic division, so evaluating p at r produces both the quotient p(x)/(x-r) and the remainder p(r); value and divisibility are two faces of the remainder theorem. Magenta is the quotient, green the value.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN