# 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
```

Difficulty: EASY
Tags: Fundamentals

Mathematics

Starter code:
def polynomialize(roots):
	

Solutions:
import re

def polynomialize(roots):

	def deploy(roots):
		r = -roots[0]
		if len(roots) == 1:
			return [r, 1]
		sub = deploy(roots[1:]) + [0]
		return [c * r + sub[i - 1] for (i, c) in enumerate(sub)]
	coefs = deploy(roots)
	poly = ' + '.join(['{}x^{}'.format(c, i) for (i, c) in enumerate(coefs) if c][::-1])
	poly = re.sub('x\\^0|\\^1\\b|\\b1(?=x)(?!x\\^0)', '', poly).replace('+ -', '- ') + ' = 0'
	return poly


def poly(lst):
	ans = [1, -lst.pop()]
	while lst:
		ans.append(0)
		r = lst.pop()
		for i in range(len(ans) - 1, 0, -1):
			ans[i] -= r * ans[i - 1]
	return ans
import re

def polynomialize(roots):
	if len(roots) == 1:
		if roots[0] == 0:
			return 'x = 0'
		elif roots[0] < 0:
			return 'x + ' + str(abs(roots[0])) + ' = 0'
		else:
			return 'x - ' + str(roots[0]) + ' = 0'
	if len([i for i in roots if i != 0]) == 0:
		return 'x^' + str(len(roots)) + ' = 0'
	ans = poly(roots)
	ans1 = ans[:]
	final = []
	for i in range(len(ans[:-1])):
		try:
			if int(ans[i]) == 0:
				continue
		except:
			continue
		if int(ans[i]) == 1:
			num = len(ans1[:-1]) - i
			if num == 1:
				final.append('x')
			else:
				final.append('x^{}'.format(str(num)))
		else:
			num = len(ans1[:-1]) - i
			if num == 1:
				final.append(str(int(ans[i])) + 'x')
			else:
				final.append(str(int(ans[i])) + 'x^{}'.format(str(num)))
	if int(ans[-1]) != 0:
		final.append(str(int(ans[-1])))
	if isinstance(final[0], list):
		final = final[-1]
	answer = '+'.join(final)
	answer = answer.replace('+-', ' - ')
	answer = answer.replace('+', ' + ')
	answer = re.sub(' 1x', ' x', answer)
	return answer + ' = 0'


def polynomialize(roots):
	coeffs = [1]
	for root in roots:
		coeffs = [1] + [coeffs[i] + coeffs[i - 1] * -root for i in range(1, len(coeffs))] + [-root * coeffs[-1]]
	coeffs = [d for d in list(enumerate(coeffs[::-1]))[::-1] if d[1] != 0]
	r = ['{:+0d}x^{}'.format(d[1], d[0]) for d in coeffs if d[0] > 1]
	r += ['{:+0d}x'.format(d[1]) for d in coeffs if d[0] == 1]
	r += ['{:+0d}'.format(d[1]) for d in coeffs if d[0] == 0]
	f = ' '.join(r).replace('+1x', '+x').replace('-1x', '-x').replace('-', '- ').replace('+', '+ ').lstrip('+').strip(' ')
	return f + ' = 0'


import numpy as np
string = lambda v: '' if not v else 'x' if v == 1 else f'x^{v}'

def coefficients(roots):
	coeffs = [1, -roots.pop()]
	while roots:
		coeffs.append(0)
		r = roots.pop()
		for i in reversed(range(1, len(coeffs))):
			coeffs[i] -= r * coeffs[i - 1]
	return coeffs

def polynomialize(roots):
	coeffs = coefficients(roots)
	z = zip(reversed(range(len(coeffs))), coeffs)
	res = [string(next(z)[0])]
	res.extend((f" {'-+'[b > 0]} {(abs(b), '')[a and abs(b) == 1]}{string(a)}" for (a, b) in z if b))
	res.append(' = 0')
	return ''.join(res)


def polynomialize(roots):
	space_sign_space = lambda c: ' + ' if 0 < c else ' - '
	space_sign_space_num = lambda c: space_sign_space(c) + (str(abs(c)) if abs(c) != 1 else '')
	max_power = len(roots)
	coefs = [1, -roots[0]]
	if max_power > 1:
		for r in roots[1:]:
			coefs = [1] + [c[0] - r * c[1] for c in zip(coefs[1:], coefs[:-1])] + [-r * coefs[-1]]
	eq = 'x' + ('^' + str(max_power) if max_power > 1 else '')
	power = max_power
	for c in coefs[1:-2]:
		power -= 1
		if c == 0:
			continue
		eq += space_sign_space_num(c)
		eq += 'x^' + str(power)
	if max_power > 1 and coefs[-2] != 0:
		eq += space_sign_space_num(coefs[-2])
		eq += 'x'
	if coefs[-1] != 0:
		eq += space_sign_space(coefs[-1]) + str(abs(coefs[-1]))
	eq += ' = 0'
	return eq


def polynomialize(roots):

	def add_root(poly, root):
		poly1 = poly + [0]
		poly2 = [0] + [-root * coef for coef in poly]
		poly = [coef1 + coef2 for (coef1, coef2) in zip(poly1, poly2)]
		return poly

	def poly2str(poly):
		spoly = ''
		for (i, coef) in enumerate(poly):
			if i == 0:
				signum = ''
			elif coef > 0:
				signum = ' + '
			elif coef < 0:
				signum = ' - '
			else:
				continue
			if abs(coef) == 1:
				scoef = ''
			else:
				scoef = str(abs(coef))
			exp = len(poly) - i - 1
			if exp == 1:
				sexp = 'x'
			elif exp == 0:
				if scoef == '':
					sexp = '1'
				else:
					sexp = ''
			else:
				sexp = 'x^' + str(exp)
			spoly += signum + scoef + sexp
		spoly += ' = 0'
		return spoly
	poly = [1, -roots[0]]
	for root in roots[1:]:
		poly = add_root(poly, root)
	return poly2str(poly)


from functools import reduce

def toTerm(coef, power):
	if not coef or coef == 0:
		return ''
	if power == 0:
		p = ''
	elif power == 1:
		p = 'x'
	else:
		p = 'x^' + str(power)
	term = ('' if abs(coef) == 1 and power > 0 else str(abs(coef))) + p
	return (' - ' if coef < 0 else ' + ') + str(term) if term else ''

def co2poly(coefs):
	terms = [toTerm(c, i) for (i, c) in enumerate(coefs)]
	terms.reverse()
	t = ''.join(terms)
	return (t[3:] if t.startswith(' +') else t[1:]) + ' = 0'

def toCoefs(coefs, root):
	return [n * -root + c for (c, n) in zip([0] + coefs, coefs + [0])]

def polynomialize(roots):
	coefs = reduce(toCoefs, roots, [1])
	return co2poly(coefs)


def monomial(factor, index):
	if factor < 0:
		if factor == -1:
			if not index:
				return ' - 1'
			return ' - x' if index == 1 else ' - x^{}'.format(index)
		else:
			if not index:
				return ' - {}'.format(abs(factor))
			return ' - {}x'.format(abs(factor)) if index == 1 else ' - {}x^{}'.format(abs(factor), index)
	elif factor == 1:
		if not index:
			return ' + 1'
		return ' + x' if index == 1 else ' + x^{}'.format(index)
	else:
		if not index:
			return ' + {}'.format(factor)
		return ' + {}x'.format(factor) if index == 1 else ' + {}x^{}'.format(factor, index)

def polynomialize(roots):
	(length, roots, parameters) = (len(roots), filter(lambda i: i, roots), [1])
	for root in roots:
		parameters = [1] + [parameters[i + 1] - root * parameters[i] for i in range(len(parameters) - 1)] + [-parameters[-1] * root]
	return ''.join([monomial(param, length - i) for (i, param) in enumerate(parameters) if param])[3:] + ' = 0'


def monomial(factor, index):
	if factor < 0:
		if factor == -1:
			if not index:
				return ' - 1'
			return ' - x' if index == 1 else ' - x^{}'.format(index)
		else:
			if not index:
				return ' - {}'.format(abs(factor))
			return ' - {}x'.format(abs(factor)) if index == 1 else ' - {}x^{}'.format(abs(factor), index)
	elif factor == 1:
		if not index:
			return ' + 1'
		return ' + x' if index == 1 else ' + x^{}'.format(index)
	else:
		if not index:
			return ' + {}'.format(factor)
		return ' + {}x'.format(factor) if index == 1 else ' + {}x^{}'.format(factor, index)

def polynomialize(roots):
	(zeros, roots2, length) = (roots.count(0), [r for r in roots if r], len(roots))
	parameters = [1]
	for root in roots2:
		parameters = [1] + [parameters[i + 1] - root * parameters[i] for i in range(len(parameters) - 1)] + [-parameters[-1] * root]
	return ''.join([monomial(param, length - i) for (i, param) in enumerate(parameters) if param])[3:] + ' = 0'


def polynomialize(roots):
	poly = [1]
	for root in roots:
		new_poly = [1]
		for i in range(1, len(poly)):
			new_poly.append(poly[i] - root * poly[i - 1])
		new_poly.append(poly[-1] * -root)
		poly = new_poly
	str_poly = []
	for (i, coeff) in enumerate(poly):
		power = len(roots) - i
		if coeff < 0:
			str_poly.append(' - ')
		elif coeff > 0:
			if i:
				str_poly.append(' + ')
		else:
			continue
		coeff = abs(coeff)
		if power == 0 or coeff != 1:
			str_poly.append(str(coeff))
		if power != 0:
			str_poly.append('x')
		if power >= 2:
			str_poly.append('^')
			str_poly.append(str(power))
	return f"{''.join(str_poly)} = 0"