Question

If $\alpha$ is a repeated root of quadratic equation $f\left(x\right)=0$ and $A\left(x\right),B\left(x\right),C\left(x\right)$ be polynomial of degree $>2$ then the determinant $\left|\begin{array}{ccc}A\left(x\right)& B\left(x\right)& C\left(x\right)\\ A\left(\alpha \right)& B\left(\alpha \right)& C\left(\alpha \right)\\ A^{\prime }\left(\alpha \right)& B^{\prime }\left(\alpha \right)& C^{\prime }\left(\alpha \right)\end{array}\right|$ is divisible by

• A. $A\left(x\right)$
• B. $B\left(x\right)$
• C. $C\left(x\right)$
• D. $f\left(x\right)$

Answer 1

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

Let $△\left(x\right)=\left|\begin{array}{ccc}A\left(x\right)& B\left(x\right)& C\left(x\right)\\ A\left(\alpha \right)& B\left(\alpha \right)& C\left(\alpha \right)\\ A\left(\alpha \right)& B\left(\alpha \right)& C\left(\alpha \right)\end{array}\right|,$ then

${△}^{\prime }\left(x\right)=\left|\begin{array}{ccc}{A}^{\prime }\left(x\right)& B^{\prime }\left(x\right)& C^{\prime }\left(x\right)\\ A^{\prime }\left(\alpha \right)& B^{\prime }\left(\alpha \right)& C^{\prime }\left(\alpha \right)\\ A^{\prime }\left(\alpha \right)& B^{\prime }\left(\alpha \right)& C^{\prime }\left(\alpha \right)\end{array}\right|$

So, $△\left(\alpha \right)=0={△}^{\prime }\left(\alpha \right),$ therefore $\alpha$ is a repeated root of $△\left(x\right)$ and $\alpha$ is a repeated root of the quadratic equation $f\left(x\right)=0,$ so $△$ is divisible by $f\left(x\right).$