Question

If $\alpha$ be a repeated root of a quadratic equation $f\left(x\right)=0$ and $p\left(x\right),q\left(x\right)$ and $r\left(x\right)$ be the polynomial function of degree $3$, then
$\left|\begin{array}{ccc}p\left(x\right)& q\left(x\right)& r\left(x\right)\\ p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\end{array}\right|$ is always divisible by

• A. $f\left(x\right)$
• B. $\alpha f\left(x\right)$
• C. $(x-\alpha )f\left(x\right)$
• D. $(x-\alpha {)}^{2}f(x)$

Answer 1

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

Since, $\alpha$ be a repeated root of a quadratic equation $f\left(x\right)=0$.
$\therefore f\left(x\right)=(x-\alpha {)}^2$

Let $\Delta \left(x\right)=\left|\begin{array}{ccc}p\left(x\right)& q\left(x\right)& r\left(x\right)\\ p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\end{array}\right|...\left(1\right)$

$\Delta \left(\alpha \right)=\left|\begin{array}{ccc}p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\end{array}\right|$

$=0$
$\therefore x-\alpha$ divides $\Delta \left(x\right)$

Differentiate eqn(1) w.r.t. $x$, we get
${\Delta }^{\prime }\left(x\right)=\left|\begin{array}{ccc}{p}^{\prime }\left(x\right)& q^{\prime }\left(x\right)& r^{\prime }\left(x\right)\\ p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\end{array}\right|$

${\Delta }^{\prime }\left(\alpha \right)=\left|\begin{array}{ccc}{p}^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\\ p\left(\alpha \right)& q\left(\alpha \right)& r\left(\alpha \right)\\ p^{\prime }\left(\alpha \right)& q^{\prime }\left(\alpha \right)& r^{\prime }\left(\alpha \right)\end{array}\right|$

$=0$
$\Rightarrow x-\alpha$ is a repeated root.
$\therefore (x-\alpha {)}^2$ divides $\Delta \left(x\right)$

Hence, $\Delta \left(x\right)$ is divisible by $f\left(x\right)$ only.