Polynomial Input Scale Transform

Description

Transform ascending coefficients from p(x) to p(scale*x); differs from output scaling and retains explicit high-power zero 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.

Transform ascending coefficients from p(x) to p(scale*x); differs from output scaling and retains explicit high-power zero coefficients.

When to use this tool

Use this tool to transform ascending coefficients from p(x) to p(scale*x); differs from output scaling and retains explicit high-power zero coefficients.

How to use the tool and how it works

Enter coefficients, scale 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.

Multiply coefficient j by scale to the power j, returning coefficients for p(scale × x).^[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.
scale	Finite argument multiplier: the output represents p(scale × x).	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.

json

{
  "input": {
    "coefficients": [
      1,
      2,
      3
    ],
    "scale": 2
  },
  "output": {
    "result": {
      "coefficients": [
        1,
        4,
        12
      ]
    }
  }
}

Limitations and assumptions

The constant coefficient is unchanged; negative scale and zero scale are allowed.
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 Input Scale Transform — tool.io

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 Translation Transform
Translate polynomial input to p(x+offset) by bounded binomial expansion of ascending coefficients; does not approximate or fit coefficients.