Description
Select one ascending polynomial power coefficient, distinguishing implicit zero beyond stored degree from an explicit supplied zero.
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.
Select one ascending polynomial power coefficient, distinguishing implicit zero beyond stored degree from an explicit supplied zero.
When to use this tool
Use this tool to select one ascending polynomial power coefficient, distinguishing implicit zero beyond stored degree from an explicit supplied zero.
How to use the tool and how it works
- Enter coefficients, power using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.coefficient, result.explicit. Use the ordering, boundary rules and absence conditions described below.
Return the coefficient at the requested nonnegative power, or zero beyond the stored list, alongside explicit indicating whether that position was stored.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. |
| power | Nonnegative integer coefficient power to inspect. | A finite safe integer within the operation’s stated bounds. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.coefficient | Stored coefficient at power, or implicit zero beyond the list. |
| result.explicit | Whether power is inside the stored coefficient list. |
{
"input": {
"coefficients": [
1,
2,
0,
0
],
"power": 5
},
"output": {
"result": {
"coefficient": 0,
"explicit": false
}
}
}Limitations and assumptions
- An explicitly stored zero and an implicit zero beyond the list are distinguished.
- 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.