Question

If $x^2-px+q;q\ne 0$, is a factor of $x^3-a{x}^2+bx-c;c\ne 0$, then

• A. $pq+c=aq$
• B. $q(ac-b+1)=c^2$
• C. $q^2+c^2=pqc$
• D. $a^2+b^2+c^2=p^2+q^2$

Answer 1

Answer: (A)

$pq+c=aq$

$x^2-px+q$ , $q\ne 0$ is a factor of $x^3-a{x}^2+bx-c$, $c\ne 0$

Let $\alpha ,\beta$ be roots of $x^2-px+q=0$ and $\alpha ,\beta ,\gamma$ be roots of $x^3-a{x}^2+bx-c=0$

$\Rightarrow \alpha +\beta =p;\alpha \beta =q;\alpha +\beta +\gamma =a$ and $\alpha \beta \gamma =c$

$\to c=\alpha \beta \gamma =q(a-p)$

$\therefore pq+c=aq$