For each of the following find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorisation. (i) `(-8)/(3),(4)/(3)` (ii) `(-3)/(2sqrt(5)),-(1)/(2)`

Explanation

(i) Given that, sum of zeroess `(S) = - (8)/(3)` and product of zeroes `(P) = (4)/(3)`
`:.` Required quadratic expression, `f(x) = x^(2) - Sx +P`
`= x^(2) +(8)/(3)x +(4)/(3) = 3x^(2) +8x +4`
Using factorisation method, `=3x^(2) +6x +2x +4`
`=3x (x+2) +2 (x+2) = (x+2) (3x+2)`
Hence, the zeroes of `f(x)` are `-2` and `-(2)/(3)`.
(ii) Given that, `S =(21)/(8)` and `P = (5)/(16)`
`:.` Required quadratic expression, `f(x) = x^(2)-Sx +P`
`= x^(2)-(21)/(8)x +(5)/(16) = 16x^(2) - 42x +5`
Using factorisation method `= 16 x^(2) - 40x - 2x +5`
`= 8x (2x -5) -1(2x -5) =(2x-5) (8x-1)`
Hence, the zeroes of `f(x)` are `(5)/(2)` and `(1)/(8)`
(iii) Given that, `S =- 2 sqrt(3)` and `P =- 9`
`:.` Required quadratic expression,
`f(x) = x^(2) -Sx +P = x^(2) +2 sqrt(3)x - 9`
`= x^(2) +3 sqrt(3)x - sqrt(3)x - 9` [using factorisation method]
`= x(x+3sqrt(3)) -sqrt(3) (x+3sqrt(3))`
`= (x + 3sqrt(3)) (x-sqrt(3))`
Hence, the zeroes of `f(x)` are `-3sqrt(3)` and `sqrt(3)`.
(iv) Given that, `S =- (3)/(2sqrt(5))` and `P =- (1)/(2)`
`:.` Required quadratic expression,
`f(x) = x^(2)-Sx +P = x^(2) + (3)/(2sqrt(5)) x - (1)/(2)`
`= 2 sqrt(5) x^(2) +3x - sqrt(5)`
using factorisation method, `=2sqrt(5) x^(2) +5x - 2x - sqrt(5)`
`= sqrt(5)x (2x +sqrt(5)) -1 (2x +sqrt(5))`
`= (2x +sqrt(5)) (sqrt(5)x-)`
Hence, the zeroes of `f(x)` are `-(sqrt(5))/(2)` and `(1)/(sqrt(5))`.