Differentiation of a Determinant
where
a dash denotes derivative with respect to x.
Example.1
Let
a be a repeated root
of the quadratic equation f(x) =0 and A(x), B(x), C(x) be polynomials of
degree 3,4, and 5 respectively. Then show that 
is
divisible by f(x), where a dash denotes the derivative, with respect to x.
Solution
Let g(x) = 
We
find that g(a)=0, & g'(a)=0
&⇒ g(x)=(x-a)2 P(x), where P(x) is a
polynomial of degree 3. Also f(x)=0 is a quadratic equation having repeated
root a .
&⇒ g(x) is divisible
by f(x).
