






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

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

				_SCREAMING_SNAKE_CASE      :		int							=						{x for x, _ in coordinates}
				if len(lowerCamelCase					) == 1:
								return f"""x={coordinates[0][0]}"""

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

				_SCREAMING_SNAKE_CASE      :		Tuple							=						len(lowerCamelCase					)

				_SCREAMING_SNAKE_CASE      :		Union[str, Any]							=						0
				_SCREAMING_SNAKE_CASE      :		list[list[float]]							=						[]
				# put the x and x to the power values in a matrix
				while count_of_line < x:
								_SCREAMING_SNAKE_CASE      :		Any							=						0
								_SCREAMING_SNAKE_CASE      :		Optional[Any]							=						coordinates[count_of_line][0]
								_SCREAMING_SNAKE_CASE      :		list[float]							=						[]
								while count_in_line < x:
												count_line.append(a ** (x - (count_in_line + 1))					)
												count_in_line += 1
								matrix.append(lowerCamelCase					)
								count_of_line += 1

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

				_SCREAMING_SNAKE_CASE      :		Tuple							=						0

				while count < x:
								_SCREAMING_SNAKE_CASE      :		List[str]							=						0
								while zahlen < x:
												if count == zahlen:
																zahlen += 1
												if zahlen == x:
																break
												_SCREAMING_SNAKE_CASE      :		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

				_SCREAMING_SNAKE_CASE      :		Optional[int]							=						0
				# make solutions
				_SCREAMING_SNAKE_CASE      :		list[str]							=						[]
				while count < x:
								solution.append(str(vector[count] / matrix[count][count]					)					)
								count += 1

				_SCREAMING_SNAKE_CASE      :		Optional[int]							=						0
				_SCREAMING_SNAKE_CASE      :		List[Any]							=						'f(x)='

				while count < x:
								_SCREAMING_SNAKE_CASE      :		list[str]							=						solution[count].split('E'					)
								if len(lowerCamelCase					) > 1:
												_SCREAMING_SNAKE_CASE      :		int							=						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, 11]]))
			print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
			print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))









"""simple docstring"""
def 							lowercase__		(				lowerCamelCase					):
				return -num if num < 0 else num





def 							lowercase__		(				lowerCamelCase					):
				if len(lowerCamelCase					) == 0:
								raise ValueError('abs_min() arg is an empty sequence'					)
				_SCREAMING_SNAKE_CASE      :		List[Any]							=						x[0]
				for i in x:
								if abs_val(lowerCamelCase					) < abs_val(lowerCamelCase					):
												_SCREAMING_SNAKE_CASE      :		List[Any]							=						i
				return j





def 							lowercase__		(				lowerCamelCase					):
				if len(lowerCamelCase					) == 0:
								raise ValueError('abs_max() arg is an empty sequence'					)
				_SCREAMING_SNAKE_CASE      :		Tuple							=						x[0]
				for i in x:
								if abs(lowerCamelCase					) > abs(lowerCamelCase					):
												_SCREAMING_SNAKE_CASE      :		Optional[Any]							=						i
				return j





def 							lowercase__		(				lowerCamelCase					):
				if len(lowerCamelCase					) == 0:
								raise ValueError('abs_max_sort() arg is an empty sequence'					)
				return sorted(lowerCamelCase,	key=lowerCamelCase					)[-1]





def 							lowercase__		(				):
				assert abs_val(0					) == 0
				assert abs_val(34					) == 34
				assert abs_val(-100_000_000_000					) == 100_000_000_000

				_SCREAMING_SNAKE_CASE      :		Tuple							=						[-3, -1, 2, -11]
				assert abs_max(lowerCamelCase					) == -11
				assert abs_max_sort(lowerCamelCase					) == -11
				assert abs_min(lowerCamelCase					) == -1


if __name__ == "__main__":
			import doctest

			doctest.testmod()

			test_abs_val()
			print(abs_val(-34))  # --> 34

