If `x^3+3x^2-9x+c` is of the form `(x-alpha)^2(x-beta)` then `c` is equal to

Options

A. 27
B. -27
C. 5
D. -5

Answer

2,3

Explanation

` f(x) = x^(3) 3x^(2) - 9x + c ` is of the form ` (x - alpha )^(2) (x - beta)` , showing
that ` alpha ` is a double root so that ` f'(x) = 0` has also one roots `alpha `.
i.e., ` 3x^(2) + 6x - 9 = 0 ` has one root ` alpha ` . Hence , ` x^(2) + 2x - 3 = 0 ` or
`(x + 3) (x - 1) = 0 ` has the root ` alpha ` which can be either - 3 or 1 .
If ` alpha = 1 ` , then f(x) = 0 given c - 5 = 0 or c = 5 . If ` alpha = - 3 ` , then
f(x) = 0 ` given
` - 27 + 27 + 27 + c = 0`
` therefore c = - 27 `