If the HCF of the polynomials $x^3+px+q $ and $x^3+rx^2+lx+x$ is $x^2+ax+b$, then their LCM is? (provided that $r≠0$) I tried multiplying the two polynomials together and then dividing them by the HCF (as product=HCF*LCM), but reached nowhere.
Then, I used the factor theorem but also got stuck.
Can somebody help?

If $b \neq 0$,
$$x^2+ax+b \mid x(x^2+rx+l+1) $$
This would imply that $x^2+ax+b = x^2+rx+l+1 $.
And the LCM would be,
$$\frac{x(x^2+rx+l+1)(x^3+px+q)}{x^2+ax+b}=x^4+px^2+q$$
If $b=0$,
$$x+a|x^2+rx+l+1$$
if you divide you'll find the quotient to be $x+(r-a)$ and remainder to be $l+1-a(r-a)=0$
LCM is,
$$(x^3+px+q)(x+r-a)$$
