Question

$f\left(x\right),g\left(x\right)$ are two polynomial with integer coefficient such that their H.C.F. is $1$ and LCM is $(x^2-4)(x^4-1)$. If $f\left(x\right)=x^3-2x^2-x+2$, then $g\left(x\right)$ is

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

Answer 1

Answer: (C)

$x^3+2x^2+x+2$

$HCF\times LCM=f\left(x\right)\times g\left(x\right)$
$g\left(x\right)=\frac{1\times (x^2-4)(x^4-1)}{x^3-2x^2-x+2},f\left(x\right)=x^3-2x^2-x+2$; $(x-1)$ is a factor of $f\left(x\right)$,
$\therefore x^3-2x^2-x+2$
$=x^2(x-1)-x(x-1)-2(x-1)$
$=(x-1)(x^2-x-2)$
$=(x-1)(x-2)(x+1)$
$\therefore g\left(x\right)=\frac{(x-2)(x+2)(x+1)(x^2+1)(x-1)}{(x-1)(x-2)(x+1)}$
$\Rightarrow g\left(x\right)=(x+2)(x^2+1)$
$\Rightarrow g\left(x\right)=x^3+2x^2+x+2$