A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The remainder when P(x) is divided by (x - 1) (x - 2) is

Given, polynomial $$P(x)$$ leaves a remainder 2 when divided by $$(x-1)$$

or, $$P(1)=2$$

similarly, $$P(x)$$ leaves a remainder 1 when divided by $$(x-2)$$

or, $$P(2)=1$$

Now, when divided by $$(x-1)(x-2)$$, which is a second degree polynomial, remainder must be linear (of the form $$Ax+B$$)

(As remainders always are a degree less than the divisor)

So, let the remainder be $$Ax+B$$, when P(x) is divided by (x-1)(x-2),

Now, $$P(1)=2$$

or, $$A+B=2$$ ----->(i)

also,$$P(2)=1$$

or, $$2A+B=1$$ ------>(ii)

Upon solving equations (i) and (ii), we get, $$A=-1,B=3$$

So the remainder is $$(-1)\cdot x+3=3-x$$