A quadratic polynomial, whose zeroes are-3 and 4, is

Options

A. `x^(2)-x +12`
B. `x^(2)+x+12`
C. `(x^(2))/(2)-(x)/(2)-6`
D. `2x^(2)+2x-24`

Answer

C

Explanation

Let `ax^(2)+bx +c` be a required polynomial whose zeroes are -3 and 4,
Then, sum of zeroes `=- 3+4 = 1" "[ :' "sum of zeroes"=(-b)/(a)]` ...(i)
`rArr (-b)/(a)=(1)/(1) rArr (-b)/(a) =- ((-1))/(1)`
and product of zeroes `=- 3 xx 4 =- 12" "[ :' "product of zeroes" =(c)/(a)]` ...(ii)
`rArr (c)/(a) =(-12)/(1)`
From Eqs. (i) and (ii),
`a = 1, b =- 1` and `c =- 12`
`=ax^(2)+bx +c`
`:.` Required polynomial `= 1.x^(2) -1.x - 12`
`= x^(2)-x - 12`
`=(x^(2))/(2)- (x)/(2)-6`
We know that, if we multiply/divide and polynomial by any constant, then the zeroes of polynomial do not change.
Altrnate Method
Let the zeroes of a quadric polynomial are `alpha =- 3` and `beta = 4`.
The, sum of zeroes `= alpha +beta =- 3 +4 =1`
and product of zeroes `= alpha beta = (-3) (4)=-12`
`:.` Required polynomial `= x^(2) -`(sum of zeroes) `x+` (product of zeroes)
`= x^(2) -(1)x +(-12) = x^(2) -x -12`
`(x^(2))/(2)-(x)/(2)-6`