Remainder and long divisionI have been thinking about this. Lets say we take 7 divided by 3, we know the remainder is 1. However, if we let x=7 and x-4=3, and we take x/(x-4), after performing long division the remainder is 4. Why is it not 1?

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ahem37 · Accepted Answer

Because the quotients are different:  7  =  2  ⋅  3  +  1  x  =  1  ⋅  (  x  −  4  )  +  4When   x  =  7 we get  7  =  1  ⋅  3  +  4which is correct, of course.

Janet Hart · Accepted Answer

If someone handed you   x and   x  −  4, why would you assume that they are   7 and   3?In general, we know that             x              x        −        4              =    1    +          4              x        −        4            For   x  =  7, it is the same that             7              7        −        4              =    1    +          4              7        −        4            , just like any arbitrary   xHowever, it just so happens that       1    +          4              7        −        4             can be rewritten as       2    +          1              7        −        4            The polynomial remainder is still   4

I have been thinking about this. Lets say we take 7 divided by 3, we know the remainder is 1. However, if we let x=7 and x-4=3, and we take x/(x-4), after performing long division the remainder is 4. Why is it not 1?

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