


def lowerCamelCase    (    lowerCAmelCase       :			Dict	):





	"""simple docstring"""







	if len(_UpperCAmelCase	) == 0 or not all(len(_UpperCAmelCase	) == 2 for pair in coordinates	):
		raise ValueError('The program cannot work out a fitting polynomial.'	)

	if len({tuple(_UpperCAmelCase	) for pair in coordinates}	) != len(_UpperCAmelCase	):
		raise ValueError('The program cannot work out a fitting polynomial.'	)

	__magic_name__ :					int  			=  {x for x, _ in coordinates}
	if len(_UpperCAmelCase	) == 1:
		return f'x={coordinates[0][0]}'

	if len(_UpperCAmelCase	) != len(_UpperCAmelCase	):
		raise ValueError('The program cannot work out a fitting polynomial.'	)

	__magic_name__ :					Any  			=  len(_UpperCAmelCase	)

	__magic_name__ :					Optional[Any]  			=  0
	__magic_name__ :					Optional[int]  			=  []
	# put the x and x to the power values in a matrix
	while count_of_line < x:
		__magic_name__ :					Tuple  			=  0
		__magic_name__ :					List[Any]  			=  coordinates[count_of_line][0]
		__magic_name__ :					Optional[int]  			=  []
		while count_in_line < x:
			count_line.append(a ** (x - (count_in_line + 1))	)
			count_in_line += 1
		matrix.append(_UpperCAmelCase	)
		count_of_line += 1

	__magic_name__ :					Optional[int]  			=  0
	# put the y values into a vector
	__magic_name__ :					str  			=  []
	while count_of_line < x:
		vector.append(coordinates[count_of_line][1]	)
		count_of_line += 1

	__magic_name__ :					str  			=  0

	while count < x:
		__magic_name__ :					Optional[Any]  			=  0
		while zahlen < x:
			if count == zahlen:
				zahlen += 1
			if zahlen == x:
				break
			__magic_name__ :					List[str]  			=  matrix[zahlen][count] / matrix[count][count]
			for counting_columns, item in enumerate(matrix[count]	):
				# manipulating all the values in the matrix
				matrix[zahlen][counting_columns] -= item * bruch
			# manipulating the values in the vector
			vector[zahlen] -= vector[count] * bruch
			zahlen += 1
		count += 1

	__magic_name__ :					str  			=  0
	# make solutions
	__magic_name__ :					Dict  			=  []
	while count < x:
		solution.append(str(vector[count] / matrix[count][count]	)	)
		count += 1

	__magic_name__ :					List[Any]  			=  0
	__magic_name__ :					Tuple  			=  'f(x)='

	while count < x:
		__magic_name__ :					Optional[int]  			=  solution[count].split('E'	)
		if len(_UpperCAmelCase	) > 1:
			__magic_name__ :					Union[str, Any]  			=  f'{remove_e[0]}*10^{remove_e[1]}'
		solved += f'x^{x - (count + 1)}*{solution[count]}'
		if count + 1 != x:
			solved += "+"
		count += 1

	return solved


if __name__ == "__main__":
							print(points_to_polynomial([]))
							print(points_to_polynomial([[]]))
							print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
							print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
							print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
							print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
							print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
							print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
							print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
							print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))



'''simple docstring'''

from __future__ import annotations





def lowerCamelCase    (    lowerCAmelCase       :			list[int | float]	,	lowerCAmelCase       :			int	,	lowerCAmelCase       :			int	):





	"""simple docstring"""







	if len(lowerCAmelCase	) == 0:
		raise ValueError('find_min() arg is an empty sequence'	)
	if (
	    left >= len(lowerCAmelCase	)
	    or left < -len(lowerCAmelCase	)
	    or right >= len(lowerCAmelCase	)
	    or right < -len(lowerCAmelCase	)
	):
		raise IndexError('list index out of range'	)
	if left == right:
		return nums[left]
	__magic_name__ :					Tuple  			=  (left + right) >> 1  # the middle
	__magic_name__ :					str  			=  find_min(lowerCAmelCase	,	lowerCAmelCase	,	lowerCAmelCase	)  # find min in range[left, mid]
	__magic_name__ :					Optional[Any]  			=  find_min(lowerCAmelCase	,	mid + 1	,	lowerCAmelCase	)  # find min in range[mid + 1, right]

	return left_min if left_min <= right_min else right_min


if __name__ == "__main__":
							import doctest

							doctest.testmod(verbose=True)