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

Let `phi(x)=|{:(A(x)" "B(x)" "C(x)),(A(alpha)" "B(alpha)" "C(alpha)),(A'(alpha)" "B'(alpha)" "C'(alpha)):}|" "`....(i)
Given that, `alpha` is repeated root of quadratic equation f(x) = 0.
`:." We must have " f(x) = (x-alpha)^(2)*g(x)`
`:." "phi'(x)=|{:(A'(x)" "B'(x)" "C'(x)),(A(alpha)" "B(alpha)" "C(alpha)),(A'(alpha)" "B'(alpha)" "C'(alpha)):}|`
`rArr" "phi'(alpha)=|{:(A'(alpha)" "B'(alpha)" "C'(alpha)),(A(alpha)" "B(alpha)" "C(alpha)),(A'(alpha)" "B'(alpha)" "C'(alpha)):}|=0`
` rArr" " x = alpha" is root of " phi'(x)`.
` rArr" " (x-alpha)" is a factor of "phi' (x) ` also.
or we can say `(x-alpha)^(2) ` is a factor of f (x).