





"""simple docstring"""




lowerCamelCase_	:		int         =   {hex(i)[2:]: i for i in range(16)}  # Use [:2] to strip off the leading '0x'





def 		UpperCAmelCase__						(	_UpperCAmelCase   ):






  """simple docstring"""





  A_    :			str									=    hex_string.strip().lower()
  if not hex_string:
    raise ValueError('Empty string was passed to the function'   )
  A_    :			Any									=    hex_string[0] == '-'
  if is_negative:
    A_    :			Optional[Any]									=    hex_string[1:]
  if not all(char in hex_table for char in hex_string   ):
    raise ValueError('Non-hexadecimal value was passed to the function'   )
  A_    :			Tuple									=    0
  for char in hex_string:
    A_    :			Union[str, Any]									=    16 * decimal_number + hex_table[char]
  return -decimal_number if is_negative else decimal_number


if __name__ == "__main__":
   from doctest import testmod

   testmod()






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

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

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

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

					lowerCamelCase__      =							len(a)

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

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

					lowerCamelCase__      =							0

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

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

					lowerCamelCase__      =							0
					lowerCamelCase__      =							"f(x)="

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

