Question

Let GCD $=x+1$ and LCM $=x^6-1$ for two given polynomials $f\left(x\right)$ and $g\left(x\right)$. Then, if $f\left(x\right)=x^3+1$. what is $g\left(x\right)$?

• A. $(x^3+1)(x-1)$
• B. $(x^3-1)(x+1)$
• C. $x$
• D. $x^2+1+x^3$

Answer 1

The correct option is B

$(x^3-1)(x+1)$

$f\left(x\right)=x^3+1$
$GCD=x+1$
$LCM=x^6-1$

We know that , $LCM\times GCD=f\left(x\right)\times g\left(x\right)$

$\Rightarrow g\left(x\right)=\frac{LCM\times GCD}{f\left(x\right)}=\frac{(x^6-1)(x+1)}{x^3+1}=\frac{(x^3+1)(x^3-1)(x+1)}{x^3+1}=(x^3-1)(x+1)$
$⟹g\left(x\right)=(x^3-1)(x+1)$