Description
Translate polynomial input to p(x+offset) by bounded binomial expansion of ascending coefficients; does not approximate or fit coefficients.
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.
Translate polynomial input to p(x+offset) by bounded binomial expansion of ascending coefficients; does not approximate or fit coefficients.
When to use this tool
Use this tool to translate polynomial input to p(x+offset) by bounded binomial expansion of ascending coefficients; does not approximate or fit coefficients.
How to use the tool and how it works
- Enter coefficients, offset 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.
Expand p(x + offset) with binomial coefficients and sum contributions for each resulting 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. |
| offset | Finite argument translation: the output represents p(x + offset). | A finite number satisfying the operation’s stated comparisons. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.coefficients | Result coefficients in ascending power order. |
{
"input": {
"coefficients": [
1,
2,
3
],
"offset": 1
},
"output": {
"result": {
"coefficients": [
6,
8,
3
]
}
}
}Limitations and assumptions
- At most 32 input coefficients are supported; this shifts the argument, not the constant term alone.
- 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 Input Negation Transform
Transform ascending coefficients from p(x) to p(-x), flipping odd powers only and preserving stored zeros.
- Polynomial Input Scale Transform
Transform ascending coefficients from p(x) to p(scale*x); differs from output scaling and retains explicit high-power zero coefficients.