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If $f(x), \mathrm{g}(x)$ and $\mathrm{h}(x)$ are three polynomials of degree 2 and
$$
\Delta(\mathrm{x})=\left|\begin{array}{ccc}
\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\
\mathrm{f}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\
\mathrm{f}^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})
\end{array}\right|
$$
then $\Delta(x)$ is a polynomial of degree
Options:
$$
\Delta(\mathrm{x})=\left|\begin{array}{ccc}
\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\
\mathrm{f}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\
\mathrm{f}^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime \prime}(\mathrm{x}) & \mathrm{h}^{\prime \prime}(\mathrm{x})
\end{array}\right|
$$
then $\Delta(x)$ is a polynomial of degree
Solution:
1960 Upvotes
Verified Answer
The correct answer is:
0
Since $f(x), g(x)$ and $h(x)$ are the polynomials of degree 2,
therefore $f^{\prime \prime}(x)=g^{\prime \prime \prime}(x)=h^{\prime \prime \prime}(x)=0$
$$
\begin{array}{c}
\text { Now, } \Delta(x)=\left|\begin{array}{lll}
f(x) & g(x) & h(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
\Rightarrow \Delta^{\prime}(x)=\left|\begin{array}{lll}
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
+\left|\begin{array}{lll}
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
+\left|\begin{array}{lll}
f(x) & g(x) & h(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
\Rightarrow \Delta^{\prime}(x)=0+0+0=0 \\
\text { constant. }
\end{array}
$$
Thus, $\Delta(x)$ is the polynom ial of degree zero.
therefore $f^{\prime \prime}(x)=g^{\prime \prime \prime}(x)=h^{\prime \prime \prime}(x)=0$
$$
\begin{array}{c}
\text { Now, } \Delta(x)=\left|\begin{array}{lll}
f(x) & g(x) & h(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
\Rightarrow \Delta^{\prime}(x)=\left|\begin{array}{lll}
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
+\left|\begin{array}{lll}
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
+\left|\begin{array}{lll}
f(x) & g(x) & h(x) \\
f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x)
\end{array}\right| \\
\Rightarrow \Delta^{\prime}(x)=0+0+0=0 \\
\text { constant. }
\end{array}
$$
Thus, $\Delta(x)$ is the polynom ial of degree zero.
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