1.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.(i) `1/4,-1` (ii) `sqrt(2),1/3` (iii) `0,sqrt(5)`(iv) `1,1`(v) `-1/4 , 1/4 `(vi) `4, 1`

Answer» (i) `1/4 = alpha + beta= -b/a`
`alpha*beta = -1= c/a`
for the equation `ax^2 + bx+c=0`
let a=4; `-b/a = 1/4`
`b=-1`
`c/4 = -1 => c=-4`
`p(x) = 4x^2 -x - 4`
(ii) `alpha+ beta= sqrt2 & alpha*beta = 1/3`
so, `-b/a = sqrt2 & c/a = 1/3`
if `a=3; -b/3 = sqrt2`
`b= -3 sqrt2`
`c/3 = 1/3 => c=1`
`p(x) = 3x^2 - 3 sqrt2 x +1`
(iii) `alpha+ beta = 0 = -b/a`
`alpha*beta = sqrt5= c/a`
if a=1; `-b/1=0; b=0`
`c/1 = sqrt5 => c= sqrt5`
`p(x) = x^2 + 0 xx x + sqrt5 `
`= x^2 + sqrt 5`
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