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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 $$(-1)\cdot x+3=3-x$$
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