Question

find a cubic polynomial whose sum,sum of products and product of zeroes are 3,-5,1​

Answer 1

$\sf\underline{Answer:}$

${x}^{3} - 3 {x}^{2} - 5x - 1$

$\sf\underline{Explanation:}$

Given, the sum of zeroes = 3

sum of products = -5 and product of zeroes = 1

Let the zeroes be alpha, beta and gamma

$\alpha + \beta + \gamma = 3$

$\alpha \beta + \beta \gamma + \gamma \alpha = - 5$

$\alpha \beta \gamma = 1$

The cubic polynomial would be given by,

${x}^{3} - {x}^{2} ( \alpha + \beta + \gamma ) + x( \alpha \beta + \beta \gamma + \alpha \gamma ) - \alpha \beta \gamma$

$= {x}^{3} - 3 {x}^{2} + ( - 5)x - 1$

$= {x}^{3} - 3 {x}^{2} - 5x - 1$

Have a good evening