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# When a polynomial f(x) is divided by ( x-1), the remainder is 5 and when it is divided by (x-2), the remainder is 7. Find the remainder when it is divided by (x-1) (x-2)................... (Using factor/remainder theorem).

Let us assume the polynomial to be $f\left( x \right)$.When $f\left( x \right)$ is divided by $\left( {x - 1} \right)$, we get the remainder as $5$.Therefore, $f\left( 1 \right) = 5$When $f\left( x \right)$ is divided by $\left( {x - 2} \right)$, we get the remainder as $7$.Therefore, $f\left( 2 \right) = 7$Now, when same polynomial $f\left( x \right)$is divided by $\left( {x - 1} \right)\left( {x - 2} \right)$, the remainder is given by:$f\left( x \right) = q\left( x \right).\left( {x - 1} \right)\left( {x - 2} \right) + r\left( x \right)$When, $x =1$$\Rightarrow f\left( 1 \right) = 0.\left( {x - 2} \right)q\left( 1 \right) + r\left( 1 \right) = 5$When, $x = 2$$\Rightarrow f\left( 2 \right) = 0.\left( {x - 1} \right)q\left( 2 \right) + r\left( 2 \right) = 7$Solving above two equations, we get remainder $r\left( x \right) = 2x + 3$