Limit Continuity And Differentiability Ques 25
- If $\alpha$ be a repeated roots of a quadratic equation $f(x)=0$ and $A(x), B(x)$ and $C(x)$ be polynomials of degree $3,4$ and $5$ respectively, then show that $\left|\begin{array}{ccc}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|$ is divisible by $f(x)$, where prime denotes the derivatives.
(1984, 4M)
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Solution: Let $\phi(x)=\left|\begin{array}{ccc}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|$
Given that, $\alpha$ is a repeated root of the quadratic equation $f(x)=0$.
$\therefore \quad $ We must have $f(x)=(x-\alpha)^2 \cdot g(x)$
$\therefore \quad \phi^{\prime}(x)=\left|\begin{array}{lll}A^{\prime}(x) & B^{\prime}(x) & C^{\prime}(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|$
$\Rightarrow \quad \phi^{\prime}(\alpha)=\left|\begin{array}{ccc}A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A(\alpha) & B(\alpha) & C(\alpha)\end{array}\right|=0$
$\Rightarrow \quad x=\alpha$ is root of $\phi^{\prime}(x)$.
$\Rightarrow \quad(x-\alpha)$ is a factor of $\phi^{\prime}(x)$ also.
or we can say $(x-\alpha)^2$ is a factor of $f(x)$.
$\Rightarrow \quad \phi(x)$ is divisible by $f(x)$.
BETA
