Let `alpha` be a repeated root of a quadratic equation `f(x)=0a n dA(x),B(x),C(x)` be polynomials of degrees 3, 4, and 5, respectively, then show that `|A(x)B(x)C(x)A(alpha)B(alpha)C(alpha)A '(alpha)B '(alpha)C '(alpha)|` is divisible by `f(x)` , where prime `(')` denotes the derivatives.

Explanation

Since `alpha` is a repeated root of the quadratic equation
`f(x) =0 f(x) ` can be written as
`f(x)= k(x) (x-a)^(2) ` where k is some nonzero constant.
`" Let " g(x)= |{:(A'(x),,B'(x),,c'(x)),(A(alpha),,B(alpha),,c(alpha)),(A'(alpha),,B'(alpha),,c'(alpha)):}|`
g(x) is divisible by f(x) if it is divisible by `(x-a)^(2) , i.e., g(alpha)=0 " and "g(alpha) =0`
As A(x) , B(x) and c(x) are polynomials of degrees 3,4 and 5 respectively deg . `g(x) ge 2`
`" Now ", g(alpha) = |{:(A'(x),,B'(x),,C(x)),(A(alpha),,B(alpha),,c(alpha)),(A'(alpha),,B'(alpha),,c'(alpha)):}|=0`
`(R_(1)" and " R_(2) " are identical ")`
`" Also "g(x)= `|{:(A(x),,B(x),,c(x)),(A(alpha),,B(alpha),,c(alpha)),(A'(alpha),,B'(alpha),,c'(alpha)):}|`
`:. g(alpha)=` |{:(A'(x),,B(x),,Cc(x)),(A(alpha),,B(alpha),,c(alpha)),(A'(alpha),,B'(alpha),,c'(alpha)):}|=0`
`(R_(1) " and" R_(3) " are identical")`
This implies that `f(x) ` divides `g(x).`