Question

Find a cubic polynomial whose zeros are `3, 1/2 and -1.`

Answer 1

Let `alpha = 3, beta = 1/2 and gamma =- 1.` Then,
`(alpha+beta+gamma)=(3+1/2-1) = 5/2,`
`(alpha beta+beta gamma +gamma alpha) = (3/2 -1/2-3)=(-4)/2 =- 2`
and `alpha beta gamma = {3 xx 1/2 xx (-1)} = (-3)/2.`
Hence, the required polynomial is
` x^(3) - (alpha+beta+gamma) x^(2) + (alpha beta+beta gamma+gamma alpha) x- alpha beta gamma = x^(3) -5/2 x^(2) -2x+3/2.`
Thus, `2x^(3) -5x^(2) -4x+3 ` is the desired polynomial.