Question

Let $F\left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\end{array}\right|.$ If $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ are the polynomials in $x$ of degree $3,$ then degree of $F^{\prime }\left(x\right)$ is

Answer 1

Let $F\left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\end{array}\right|$
Differentiating both sides w.r.t. $x$
$F^{\prime }\left(x\right)=\left|\begin{array}{ccc}{f}^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\end{array}\right|+\left|\begin{array}{ccc}{f}^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\end{array}\right|+\left|\begin{array}{ccc}{f}^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime \prime \prime }& g^{\prime \prime \prime \prime }& h^{\prime \prime \prime \prime }\end{array}\right|$
If any two rows or any two columns in a determinant are identical (or proportional), then the value of the determinant is zero.
$\because f,g,h$ are of degree $3$
$\Rightarrow f^{\prime \prime \prime \prime },g^{\prime \prime \prime \prime },h^{\prime \prime \prime \prime }=0$
$\therefore F^{\prime }\left(x\right)=0$
Hence, the degree of $F^{\prime }\left(x\right)$ is $0.$