Description
Multiply an ascending coefficient polynomial by x^shift using exact zero padding; nonnegative shifts only, bounded stored vector size.
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.
Multiply an ascending coefficient polynomial by x^shift using exact zero padding; nonnegative shifts only, bounded stored vector size.
When to use this tool
Use this tool to multiply an ascending coefficient polynomial by x^shift using exact zero padding; nonnegative shifts only, bounded stored vector size.
How to use the tool and how it works
- Enter coefficients, shift 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.
Prepend shift zeros to the constant-first coefficient list, representing multiplication by x to that nonnegative integer power.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. |
| shift | Nonnegative integer power of x multiplying the polynomial. | 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
],
"shift": 2
},
"output": {
"result": {
"coefficients": [
0,
0,
1,
2
]
}
}
}Limitations and assumptions
- The resulting list must fit 128 coefficients; existing high-power zeros are 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
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.