Question

If $f(x), g(x)$ and $h(x)$ are three polynomials of degree 3 then $\phi(x)=\begin{vmatrix} f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\ f^{\prime \prime \prime}(x) & g^{\prime \prime \prime}(x) & h^{\prime \prime \prime}(x) \end{vmatrix}$is a polynomial of degree

• A. 3
• B. 4
• C. 5
• D. none of these

Answer 1

Answer: (D)

We have
$\phi^{\prime}(x)=\begin{vmatrix}f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\f^{\prime \prime \prime}(x) & g^{\prime \prime \prime}(x) & h^{\prime \prime \prime}(x)\end{vmatrix}+\begin{vmatrix}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\f^{\prime \prime \prime}(x) & g^{\prime \prime \prime}(x) & h^{\prime \prime \prime}(x) \\f^{\prime \prime \prime}(x) & g^{\prime \prime \prime}(x) & h^{\prime \prime \prime}(x)\end{vmatrix}+\begin{vmatrix}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\f^{i v}(x) & g^{i v}(x) & h^v(x)\end{vmatrix}$
$=0+0+\begin{vmatrix}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\
f^{\prime \prime}(x) & g^{\prime \prime}(x) & h^{\prime \prime}(x) \\0 & 0 & 0
\end{vmatrix}=0 $
since $f, g, h$ are polynomials of degree $3, f^{i v}(x)=g^{i v}(x)=$ $h^{i v}(x)=0$
$\Rightarrow \phi(x)$ must be a constant.