Question

Let $a,b,p,q$ $\in Q$ and suppose that $f\left(x\right)=x^2+ax+b=0$ and $g\left(x\right)=x^3+px+q=0$ have a common irrational root, then

• A. $f\left(x\right)$ divides $g\left(x\right)$
• B. $g\left(x\right)\equiv xf\left(x\right)$
• C. $g\left(x\right)\equiv (x-b-q)f\left(x\right)$
• D. $g\left(x\right)=x^{2}f\left(x\right)+1$

Answer 1

The correct option is A $f\left(x\right)$ divides $g\left(x\right)$

$f\left(x\right)=0=x^2+ax+b$
$g\left(x\right)=0=x^3+px+q$
Since $\alpha$ is root so ${\alpha }^2=-(a\alpha +b)$
Substitute this in ${\alpha }^3+p\alpha +q=0$
$(a^2-b+p)\alpha +ab+q=0$
Since $a,b,p,q\in Q$, so $p=-a^2+b;q=-ab$.
$g\left(x\right)=(x-a)f\left(x\right)$.