*** Nova polynomial from roots*** 

This kata is from a series on polynomial handling. ( [#1](http://www.codewars.com/kata/nova-polynomial-1-add-1)   [#2](http://www.codewars.com/kata/570eb07e127ad107270005fe)  [#3](http://www.codewars.com/kata/5714041e8807940ff3001140 )   [#4](http://www.codewars.com/kata/571a2e2df24bdfd4e20001f5))

Consider a polynomial in a list where each element in the list element corresponds to the factors. The factor order is the position in the list. The first element is the zero order factor (the constant).

p = [a0, a1, a2, a3] signifies the polynomial a0 + a1x + a2x^2 + a3*x^3

In this kata create the polynomial from a list of roots:

[r0, r1 ,r2, r3 ]

p = (x-r0)(x-r1)(x-r2)(x-r3)

note: no roots should return the identity polynomial.

```python 
poly_from_roots([4]) = [-4, 1]
poly_from_roots([0, 0, 0, 0] ) = [0, 0, 0, 0, 1]
poly_from_roots([]) = [1]
```
The first katas of this series is preloaded in the code and can be used: [poly_add](http://www.codewars.com/kata/570eb07e127ad107270005fe) [poly_multiply](http://www.codewars.com/kata/570eb07e127ad107270005fe)

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'def poly_multiply(p1, p2):\n    answer = [0] * (len(p1) + len(p2) - 1)\n    for i, coeff1 in enumerate(p1):\n        for j, coeff2 in enumerate(p2):\n            answer[i + j] += coeff1 * coeff2\n    return [] if all(coeff == 0 for coeff in answer) else answer\n\ndef poly_from_roots(r):\n    if not r:\n        return [1]\n    parts = [[-root, 1] for root in r]\n    answer = parts[0]\n    for part in parts[1:]:\n        answer = poly_multiply(answer, part)\n    return answer'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'poly_from_roots'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 5526