Question

$f\left(x\right),g\left(x\right)$ are two polynomial with integer coefficient such that their HCF 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

The correct option is C $x^3+2x^2+x+2$

Product of the polynomials $=$ Product of their LCM and HCF
$\therefore f\left(x\right).g\left(x\right)=LCM\times HCF$
or $g\left(x\right)=\frac{1.(x^2-4)(x^4-1)}{x^3-2x^2-x+2}$
$=\frac{(x-2)(x+2)(x-1)(x+1)(x^2+1)}{x^2(x-2)-(x-2)}$
$=\frac{(x-2)(x+2)(x-1)(x+1)(x^2+1)}{(x-2)(x-1)(x+1)}$
$=(x+2)(x^2+1)$
$=x^3+2x^2+x+2$