Question

If $f\left(x\right),g\left(x\right),h\left(x\right)$ are polynomials in $x$ of degree $2$ and $F\left(x\right)=\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\end{array}\right|,$ then $F^{\prime }\left(x\right)$ is equal to

Answer 1

$F\left(x\right)=\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\end{array}\right|F^{\prime }\left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\end{array}\right|+\left|\begin{array}{ccc}f& g& h\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\\ f^{\prime \prime }& g^{\prime \prime }& h^{\prime \prime }\end{array}\right|+\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ f^{\prime \prime \prime }& g^{\prime \prime \prime }& h^{\prime \prime \prime }\end{array}\right|$
If any two rows or any two columns in a determinant are identical (or proportional), then the value of the determinant is zero.
Since $f,g,h$ are of degree $2,$
$\Rightarrow f^{\prime \prime \prime },g^{\prime \prime \prime },h^{\prime \prime \prime }=0$
$\therefore F^{\prime }\left(x\right)=0$