>>11570885A polynomial time algorithm is on that, when you give it some input, the time it takes to spit out the output grows with a function that is a polynomial with respect to the size of the input.
All these algorithms are "polynomial time", because the amount of time it takes to spit out the answer is polynomial-proportioned to the input. So if you have an algorithm that's O(n^3) time and give in an input of length 120, then the output will take 120*120*120 = 1,728,000 steps to complete. Examples of poly-time algorithms are O(logn), O(n), O(nlogn), O(n^c) where c is a constant, etc.
Now, an exponential time algorithm is one who's amount of steps take an an exponential form based on the input rather than a polynomial form. For example, O(2^n) or O(4^n). If you give an algorithm with the time complexity of O(2^n) an input of length 120, then it will take 2^120 steps to complete, which is so large that you basically will never get the output, the universe will die before the computer spits out an answer.
Now there are problems in math that, when we DON'T know what the answer is, we can show that it will take an exponential amount of time to get the answer. However, when we DO know what the answer is, it only takes a polynomial amount of time to check the answer to see if it's correct, that is, it only takes n^2 steps or something. Take sudoku. To solve sudoku takes an exponential amount of time based on the size of the board, but to CHECK to see if an answer is correct is easy - just read the board and see if it all lines up! Thus SOLVING and UNKNOWN sudoku board is expontially hard, but CHECKING an already solved sudoku puzzle is only polynomial hard - there is no symmetry there!
Then P vs. NP is asking; "Are problems who are easy to check but hard to solve, actually easy to solve as well?"
