






def        _lowerCamelCase(      lowerCAmelCase__					:		list[list[int]]   ):





				'''simple docstring'''







				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_ :						Optional[Any]    				= {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_ :						Dict    				= len(lowerCAmelCase__   )

				SCREAMING_SNAKE_CASE_ :						List[str]    				= 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_ :						str    				= 0
								SCREAMING_SNAKE_CASE_ :						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_ :						Dict    				= 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_ :						List[str]    				= 0

				while count < x:
								SCREAMING_SNAKE_CASE_ :						Optional[Any]    				= 0
								while zahlen < x:
												if count == zahlen:
																zahlen += 1
												if zahlen == x:
																break
												SCREAMING_SNAKE_CASE_ :						Any    				= 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_ :						Union[str, Any]    				= 0
				# make solutions
				SCREAMING_SNAKE_CASE_ :						list[str]    				= []
				while count < x:
								solution.append(str(vector[count] / matrix[count][count]   )   )
								count += 1

				SCREAMING_SNAKE_CASE_ :						Dict    				= 0
				SCREAMING_SNAKE_CASE_ :						Any    				= 'f(x)='

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







def        _lowerCamelCase(      lowerCAmelCase__					:		float     ,     lowerCAmelCase__					:		float     ,     lowerCAmelCase__					:		float   ):





				'''simple docstring'''







				if mass < 0:
								raise ValueError('The mass of the body cannot be negative'   )
				if radius <= 0:
								raise ValueError('The radius is always a positive non zero integer'   )
				return (mass * (velocity) ** 2) / radius


if __name__ == "__main__":
				import doctest

				doctest.testmod(verbose=True)