When an unknown polynomial is divided by (x−1) and (x−2) we obtain the remainder 2 and 1 , respectively.  Then the remainder resulting from the division of this  polynomial by (x−1)(x−2) is

Let P(x) be the unknown polynomial and let q(x) and r(x)=ax+b be the quotient and the remainder, respectively,  resulting from the division of that polynomial by (x−1)(x−2) . ⇒ P(x)=(x−1)(x−2)q(x)+ax+b from the problem P(1)=2⇒a+b=2 Also P(2)=1⇒2a+b=1⇒a=−1,b=3∴ Required remainder =(3−x)