Expert: Scotto - 6/13/2011

Question
If f(x)/(x-1)the reminder is 5 and when divided by (x-2) the remainder is 7.What is the remainder when divided by (x-2)(x-1)

Answer
Since (7-5)/[(x-1)-(x-2)] = 2/1, 2 is the number of times it divides.
Since we are dividinbg by x-2 and getting 7, x-2 must be at least 7, so x must be at least 9.
This makes the f(x) = 2x + 3.

This can be seen in the following table with a column for x, x-1, the 1st remainder, the number,
x-2, the second remainder, and the number.  Notice the number is always the same in both cases.

x, x-1, rem, number, x-2, rem, number
10,  9, 5, 23,  8, 7, 23
11, 10, 5, 25,  9   , 7, 25
12, 11, 5, 27, 10   , 7, 27
13, 12, 5, 29, 11   , 7, 29
14, 13, 5, 31, 12   , 7, 31
15, 14, 5, 33, 13   , 7, 33

For example, in row 1, x = 10, so x-1 = 9 and x-2 = 8.
This allows for the remainder on the 2nd problem to be 7.
If the number is 23, the remainder when divided by 9 { (x-1) } is 5 and
the remainder when divided by 8 { (x-2) } is 7.

If we added on (x-1)(x-2), the same principle should work.
If x=10, then (x-1)(x-2) = 9*8 = 72, and 23 + 72 = 95.
When dividing by x-1, since x=10, that is dividing by 9, and 95 divided by 9 has a remainder of 5 and
when 95 divided by by 8 { (x-2) }, the remainder is 7.

This makes the number 2x + 3 + (x-1)(x-2) = 2x + 3  +  x^2 - 3x + 2 = x^2 - x + 5.
It could also be 2x + 3  +  2(x-1)(x-2) = 2x + 3 + 2x^2 - 6x + 4 = 2x^2 - 4x + 7

In general, it could be 2x + 3 + m(x-1)(x-2) for any integers n and m.