Description
Return the first two synthetic-deflation quotient coefficients.
Polynomial coefficients encode powers in a chosen order, with zeros, multiplicities and leading terms carrying different meanings. Local coefficient identities and necessary divisibility conditions can be checked without finding every root or proving a proposed factorization. Floating-point comparisons use the exact supplied values unless an explicit tolerance is provided.
Return the first two synthetic-deflation quotient coefficients.
When to use this tool
Use Polynomial Deflation Leading Pair when checking the stated coefficient, root-identity, index or divisibility term.
How to use the tool and how it works
- Enter leading, next, root. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read quotientLeading, quotientNext. Use the output convention below.
Return the first two synthetic-deflation quotient coefficients. Require leading != 0; quotientLeading = leading, quotientNext = next+root*leading.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| leading | Leading | Required. Finite number. |
| next | Next | Required. Finite number. |
| root | Root | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| quotientLeading | Quotient Leading | Required. Finite number. |
| quotientNext | next+root*leading from one synthetic-deflation step; no complete division/remainder check. | Required. Finite number. |
{
"input": {
"leading": 2,
"next": 3,
"root": 4
},
"output": {
"quotientLeading": 2,
"quotientNext": 11
}
}Limitations and assumptions
- The proposed root and polynomial degree are supplied; this partial step does not verify zero remainder or compute a complete quotient.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a complete symbolic or numerically appropriate polynomial method for root finding, full factorization and robust treatment of near-degenerate coefficients.
References
Similar or alternative tools
- Algebra: Polynomial Leading Nonzero Coefficient
Return leading coefficient and effective degree after shared normalization.
- Polynomial Derivative Constant Coefficient
Return a polynomial derivative's constant coefficient.