# Task

Given an array of roots of a polynomial equation, you should reconstruct this equation.

___

## Output details:

* If the power equals `1`, omit it: `x = 0` instead of `x^1 = 0`
* If the power equals `0`, omit the `x`: `x - 2 = 0` instead of `x - 2x^0 = 0`
* There should be no 2 signs in a row: `x - 1 = 0` instead of `x + -1 = 0`
* If the coefficient equals `0`, skip it: `x^2 - 1 = 0` instead of `x^2 + 0x - 1 = 0`
* Repeating roots should not be filtered out: `x^2 - 4x + 4 = 0` instead of `x - 2 = 0`
* The coefficient before `q^n` is always `1`: `x^n + ... = 0` instead of `Ax^n + ... = 0`

___

## Example:

```
polynomialize([0])        =>  "x = 0"
polynomialize([1])        =>  "x - 1 = 0"
polynomialize([1, -1])    =>  "x^2 - 1 = 0"
polynomialize([0, 2, 3])  =>  "x^3 - 5x^2 + 6x = 0"
```

___

## Tests:

```python
      Main suite:                  Edge cases:
 (Reference - 4000 ms)         (Reference - 30 ms)

N of roots | N of tests      N of roots | N of tests
-----------------------      -----------------------
    1-10   |   100              20-40   |    125
 700-750   |    25               2-20   |    125
```

```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 polynomialize(roots):\n    space_sign_space = lambda c: ' + ' if 0 < c else ' - '\n    space_sign_space_num = lambda c: space_sign_space(c)  + (str(abs(c)) if abs(c) != 1 else '')\n    max_power = len(roots)\n    # Calculate coefficients\n    coefs = [1, -roots[0]]\n    if max_power > 1:\n        for r in roots[1:]:\n            coefs = [1,] + [c[0]-r*c[1] for c in zip(coefs[1:], coefs[:-1])] + [-r*coefs[-1],]\n    # Construct equation (first line, separately as it needs no leading +)\n    eq = 'x' + ('^' + str(max_power) if max_power > 1 else '')\n    power = max_power\n    # Loop for x^(max_power -1) up to x^2 if it exists\n    for c in coefs[1:-2]:\n        power -= 1\n        if c == 0:\n            continue\n        eq += space_sign_space_num(c)\n        eq += 'x^' + str(power)\n    # Coefficient for x\n    if (max_power > 1) and coefs[-2] != 0:\n        eq += space_sign_space_num(coefs[-2])\n        eq += 'x'\n    # Coefficient for const\n    if coefs[-1] != 0:\n        eq += space_sign_space(coefs[-1]) + str(abs(coefs[-1]))\n    eq += ' = 0'\n    return eq"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'polynomialize'
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: 6022