Question

If a polynomial gives remainder 1 and 2 on dividing $x-2$ and $x-1$ respectively, then the remainder when that polynomial is divided by $x^2-3x+2$ is-

• A. $x-3$
• B. $-x+3$
• C. $x-2$
• D. $-x+2$

Answer 1

The correct option is B $-x+3$

Let we assume that the polynomial is

$f\left(x\right)=(x-1)(x-2)h\left(x\right)+px+q$

Now If we divide $f\left(x\right)$ by $(x-1)$ we get remainder $2$ that is $f\left(1\right)=2$

Because $f\left(x\right)=(x-1)\alpha \left(x\right)+2$

$\Rightarrow f\left(1\right)=(1-1)\alpha \left(x\right)+2=2$

So

$f\left(1\right)=(1–1)(1–2)h\left(1\right)+p+q=2$

$\Rightarrow p+q=\mathrm{2......}\left(1\right)$

Similarly, if we divide $f\left(x\right)$ by $(x-2)$ we get remainder $1$ that is $f\left(2\right)=1$

So

$f\left(2\right)=(2–1)(2–2)h\left(2\right)+2p+q=1$

$\Rightarrow 2p+q=\mathrm{1.....}\left(2\right)$

Now subtracting (1) from (2) we get

$2p+q-(p+q)=1-2$

Therefore, $p=-1$

and

$q=2-p=2-(-1)=3$

Thus, $f\left(x\right)=(x-1)(x-2)h\left(x\right)-x+3$

So, if we divide $f\left(x\right)$ by $x^2-3x+2=(x-1)(x-2)$, the remainder $R$ will be $-x+3$.

Hence, the remainder is $-x+3$.