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

Options

A. -5
B. 27
C. -27
D. 0

Answer

C

Explanation

If `f(x) - x^(3) + 3x^(2) - 9x + c = (x - alpha)^(2) (x - beta)`, then `(x - alpha)` is a factor of order 2. So, `x - alpha` is a factor of order one of f'(x) i.e. `3x^(2) + 6x - 9`.
Now, `f'(x) = 3x^(2) + 6x - 9 = 3(x^(2) + 2x - 3)=3(x+3)(x-1)`
`therefore" "f'(x) = 0 rArr x = 1 or, x = - 3`.
This shows that either `alpha = 1 or, alpha = - 3`
But, `alpha` is a root of `x^(3) + 3x^(2) - 9x + c = 0`
`therefore" "alpha^(3) + 3 alpha^(2) - 9 alpha + c = 0 rArr 1 + 3 - 9 + c = 0" "["Putting" alpha = 1]`
and, `-27 + 27 + 27 + c = 0 rArr c = 5, -27" "["Putting" alpha = -3]`