SOLUTION: Here is my question: when p(x) is divided by x-1 the remainder is 1, and when divided by (x-2)(x-3) the remainder is 5. find remainder when P(X) is divided by (x-1)(x-2)(x-3).

Question 758299: Here is my question:
when p(x) is divided by x-1 the remainder is 1, and when divided by (x-2)(x-3) the remainder is 5. find remainder when P(X) is divided by (x-1)(x-2)(x-3).
here is part of my work:
let the quotient be Q(x), and let the remainder be ax+b
P(x)=(x-1)(x-2)(x-3)Q(X)+ax+b
P(1)=a*1+b=1 then a+b=1 (1)
then I don't know how to do with (x-2)(x-3) and the remainder is 5. Please help!
thanks

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
is cubic (degree=3) polynomial.
When you divide by the remainder polynomial could have degree 2, 1, or 0, so you should start with for a remainder.

That gives you the sytem

--> and
--> , so
-->
Then,
--> --> -->
--> -->
and finally
--> --> -->
The remainder when is divided by is

There are infinite possibilities for , as there are for . simplest corresponds to
In that case,

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