Description
Pad high powers of an ascending coefficient list to at least an explicit degree without truncating existing high powers; bounded vector output.
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.
Pad high powers of an ascending coefficient list to at least an explicit degree without truncating existing high powers; bounded vector output.
When to use this tool
Use this tool to pad high powers of an ascending coefficient list to at least an explicit degree without truncating existing high powers; bounded vector output.
How to use the tool and how it works
- Enter coefficients, degree using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.coefficients. Use the ordering, boundary rules and absence conditions described below.
Append zero coefficients until the list can represent the requested degree, leaving an already longer list unchanged.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. |
| degree | Nonnegative degree used for the requested output extent or truncation. | A finite safe integer within the operation’s stated bounds. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.coefficients | Result coefficients in ascending power order. |
{
"input": {
"coefficients": [
1,
2,
0,
0
],
"degree": 5
},
"output": {
"result": {
"coefficients": [
1,
2,
0,
0,
0,
0
]
}
}
}Limitations and assumptions
- This never truncates a polynomial and does not change its mathematical degree.
- 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
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 Low Power Remainder
Split ascending polynomial coefficients into remainder modulo x^k and quotient by x^k; not general polynomial long division.