Question

Let $f\left(x\right)$ be a polynomial with degree greater than $2$ for which remainders when divided by $(x-1),(x-2),(x-3)$ are $3,7,13$ respectively, then the remainder of $f\left(x\right)$ when divided by $(x-1)(x-2)(x-3)$, is:

• A. $2x+1$
• B. $x^2+x+1$
• C. $x^2+1$
• D. $x+2$

Answer 1

The correct option is B $x^2+x+1$

Let $f\left(x\right)=(x-1)(x-2)(x-3)g\left(x\right)+r{x}^2+sx+t$
$f\left(1\right)=(1-1)(1-2)(1-3)g\left(1\right)+r\times 1^2+s\times 1+t\Rightarrow r+s+t=3$ ...(1)
$f\left(2\right)=(2-1)(2-2)(2-3)g\left(2\right)+r\times 2^2+s\times 2+t\Rightarrow 4r+2s+t=7$ ...(2)
$f\left(3\right)=(3-1)(3-2)(3-3)g\left(3\right)+r\times 3^2+s\times 3+t\Rightarrow 9r+3s+t=13$ ...(3)
Solving (1), (2) and (3), we get
$r=1,s=1,t=1$
Hence remainder is $x^2+x+1$, when $f\left(x\right)$ is divided by $(x-1)(x-2)(x-3)$