Question

find all zeroes of the polynomial 2x^3+x^2-6x-3 if two of its zeroes are √3 and -√3

Answer 1

$p(x) = 2 {x}^{3} + {x}^{2} - 6x - 3 \\ = {x}^{2} (2x + 1) - 3(2x + 1) \\ = (2x + 1)( {x}^{2} - 3) \\ if \: p(x) = 0 \: then \\ (2x + 1) = 0 \: and \: ({x}^{2} - 3) = 0 \: which \: yields \\ (2x = - 1 \: and \: {x}^{2} = 3 \\ x = - \frac{1}{2} \: and \: x = + 3 \: \: \: x = - 3 \\ thus \: the \: other \: zero \: is \: x = - \frac{1}{2} .$