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 $(x - 1) (x - 2)$ .

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Solution

Using Division Algorithm here:-
$D i v i d e n d = D i v i s o r \times Q u o t i e n t + R e m a i n d e r$

So, Applying it $: -$
Let $q (x), k (x)$ be quotient when $f (x)$ is divided by $x - 1$ and $x - 2$ respectively

$\Rightarrow f (x) = (x - 1) q (x) + 5$
$∴ f (1) = 5$ ..... $(1)$

Also, $f (x) = (x - 2) k (x) + 7$
$∴ f (2) = 7$ ..... $(2)$

Now, let $a x + b$ be remainder when $f (x)$ is divided by $(x - 1) (x - 2)$ and $g (x)$ be quotient.
$f (x) = (x - 1) (x - 2) g (x) + (a x + b)$
Using $(1)$ and $(2)$
$5 = a + b$ ...... $(3)$
$7 = 2 a + b$ ...... $(4)$

Solving $(3)$ and $(4)$ , we get
$a = 2$ and $b = 3$
$∴ 2 x + 3$ is remainder when $f (x)$ is divided by $(x - 1) (x - 2) .$

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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 $(x - 1) (x - 2)$ .