Why does the remainder theorem work?


f (x) divided by x – a gives a remainder of f (a)

I have solved problems about this in class and know how it works, for lack of a better term. But I don’t know *why* it works or how it came to be. I know there has to be an explanation, but I can’t figure out what. I’m sorry if this is a stupid question.

In: 10

The easiest way to explain is with an example.
Let’s take 11 marbles and you need to divide it into 4 equal parts – maybe you have 4 friends you want to split them between.
Now we know 11/4 is 2.75 – so each person gets 2.75 marbles. But you can’t really have 0.75 of a marble.
So in this case everyone has 2 each and then you have a remainder of 3, to do something with, whatever that is.
Or it can make it easier find the actual decimal answer. Diving a multiple of a number is easier than if it is not a multiple. So 11 splits into – 4, then add another 4, then 3. So we have 2 lots of 4 + 3 which is not quite 4. And then we only have the remainder of 3/4 which is easier to think about than 11/4. Add it all up, 2 + 0.75 = 2.75
If we take 26/9 you might not know instinctively what that is. So let’s take the last method.
9 = 1×9, 18 = 2×9, 27 = 3×9. But we only have 26. So we have 2×9 remainder 8.
8/9 is easier than 26/9. 8/9 = 0.888888
So we add it all up. 2 + 0.8888888 = 2.888888.