Question

The GCD and LCM of two polynomials are $x+1$ and $x^6-1$ respectively. If one of the polynomials is $x^3+1$, find the other.

Answer 1

Given GCD $=x+1$ and LCM $=x^6-1$
Let $f\left(x\right)=x^3+1$.
We known that $LCM\times GCD=f\left(x\right)\times g\left(x\right)$
$\to g\left(x\right)=\frac{LCM\times GCD}{f\left(x\right)}=\frac{(x^6-1)(x+1)}{x^3+1}$
$\frac{(x^3+1)(x^3-1)(x+1)}{x^3+1}=(x^3-1)(x+1)$
Hence, $g\left(x\right)=(x^3-1)(x+1)$.