Polynomial Low Power Remainder

Description

Split ascending polynomial coefficients into remainder modulo x^k and quotient by x^k; not general polynomial long division.

A coefficient list here is constant-first: index j multiplies x to power j. Thus [1, 2, 3] represents 1 + 2x + 3x². Trailing list zeros are high-power zeros; stored length can differ from mathematical degree, and the zero polynomial needs a separate degree convention.

Split ascending polynomial coefficients into remainder modulo x^k and quotient by x^k; not general polynomial long division.

When to use this tool

Use this tool to split ascending polynomial coefficients into remainder modulo x^k and quotient by x^k; not general polynomial long division.

How to use the tool and how it works

Enter coefficients, modulusPower using the field definitions below.
Run the operation. Correct input errors before interpreting an output.
Read result.remainder, result.quotient. Use the ordering, boundary rules and absence conditions described below.

Split at modulusPower: lower coefficients form the remainder modulo x to that power, while higher coefficients form the quotient with powers shifted down.^[1]

Input field	Meaning and units	Accepted input
coefficients	Nonempty coefficients in ascending power order [constant, x, x², ...].	Dense list of at most 128 finite numbers; integer/shape constraints above also apply.
modulusPower	Positive integer m defining remainder modulo x to power m.	A finite safe integer within the operation’s stated bounds.

Output field	Interpretation
result	Object containing the named fields below.
result.remainder	Low-power coefficients of the remainder modulo x to modulusPower.
result.quotient	Higher-power coefficients shifted down by modulusPower, or [0].

json

{
  "input": {
    "coefficients": [
      1,
      2,
      3
    ],
    "modulusPower": 2
  },
  "output": {
    "result": {
      "remainder": [
        1,
        2
      ],
      "quotient": [
        3
      ]
    }
  }
}

Limitations and assumptions

If no higher coefficients exist, quotient is [0]; high-power zeros are otherwise retained.
Coefficient lists use ascending powers and ordinary floating-point arithmetic. Nonempty input is required; individual operations impose tighter size limits. The tool does not attach or check physical units for x and its coefficients.
Every declared top-level input field is required and unexpected fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
Numeric outputs must be finite and no larger in absolute magnitude than the safe-integer limit; operations returning exact large integers use decimal text. Positions are zero-based and spans end exclusively unless the operation explicitly states a different convention.

Alternative or Complementary methods

Check coefficient order before importing data from another algebra library. Use separate root-finding or symbolic algebra methods when an operation requires more than the stated coefficient transformation.

References

Tool behavior and example: Polynomial Low Power Remainder — tool.io

Similar or alternative tools

Polynomial Dense from Power Pairs
Build bounded dense ascending coefficients by summing explicit sparse power occurrences into a supplied maximum degree; duplicates are summed, missing powers are zeros.
Polynomial Coefficient Power Padding
Pad high powers of an ascending coefficient list to at least an explicit degree without truncating existing high powers; bounded vector output.