




def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							list[list[int]] )     -> str:




				'''simple docstring'''





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

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

				A__            =     {x for x, _ in coordinates}
				if len(SCREAMING_SNAKE_CASE_ ) == 1:
								return F'x={coordinates[0][0]}'

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

				A__            =     len(SCREAMING_SNAKE_CASE_ )

				A__            =     0
				A__            =     []
				# put the x and x to the power values in a matrix
				while count_of_line < x:
								A__            =     0
								A__            =     coordinates[count_of_line][0]
								A__            =     []
								while count_in_line < x:
												count_line.append(a ** (x - (count_in_line + 1)) )
												count_in_line += 1
								matrix.append(SCREAMING_SNAKE_CASE_ )
								count_of_line += 1

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

				A__            =     0

				while count < x:
								A__            =     0
								while zahlen < x:
												if count == zahlen:
																zahlen += 1
												if zahlen == x:
																break
												A__            =     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

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

				A__            =     0
				A__            =     "f(x)="

				while count < x:
								A__            =     solution[count].split("E" )
								if len(SCREAMING_SNAKE_CASE_ ) > 1:
												A__            =     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]]))




def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							list		,    SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int )     -> int:




				'''simple docstring'''





				A__            =     int((lower_bound + upper_bound) // 2 )
				if array[r] == value:
								return r
				if lower_bound >= upper_bound:
								return -1
				if array[r] < value:
								return binary_search(SCREAMING_SNAKE_CASE_		,    r + 1		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )
				else:
								return binary_search(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    r - 1		,    SCREAMING_SNAKE_CASE_ )



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							list )     -> list:




				'''simple docstring'''





				A__            =     0
				if matrix[index][0] == value:
								return [index, 0]
				while index < len(SCREAMING_SNAKE_CASE_ ) and matrix[index][0] < value:
								A__            =     binary_search(matrix[index]		,    0		,    len(matrix[index] ) - 1		,    SCREAMING_SNAKE_CASE_ )
								if r != -1:
												return [index, r]
								index += 1
				return [-1, -1]


if __name__ == "__main__":
	import doctest

	doctest.testmod()

