Question

obtain all the zeroes of polynomial f(x) =2x3-7x2+3x-6, if it's two zeroes are-√3/2, √3/2​

Answer 1

$\large \bold \red{ \underline{ \underline{answer}}} \\ \\ \bold \green{ \underline{ \underline{all \: \: zeroes \: \: are \: \: - \sqrt { \frac{3}{2} } \: \: and \: \: \sqrt{ \frac{3}{2} } \: \: and \: \: 2 \: \: and \: \: - 1 }}}$

$\large \bold \red{ \underline{ \underline{correct \: \: question}}} \\ \large \bold{find \: \: all \: \: the \: \: zeroes \: \: of \: \: the \: \: polynomial \: \: f(x) \: \: 2x {}^{4} - 2x {}^{3} - 7x {}^{2} + 3x + 6 } \: \: \large \bold{if \: two \: zeroes \: are \: \: - \sqrt{ \frac{3}{2} } \: \: and \: \: \sqrt{ \frac{3}{2} } } \\ \large \bold \blue{ \underline{ \underline{to \: find \: all \: the \: zeroes}}} \\ \large \bold \orange{ \underline{ \underline{explanation}}} \\ \large \bold{x = - \sqrt{ \frac{3}{2} } \: \: and \: \: x = \sqrt{ \frac{3}{2} } } \\ \\ \large \bold({x + \sqrt{ \frac{3}{2} } )(x - \sqrt{ \frac{3}{2} } ) = 0 } \\ \\ \large \bold{ {x}^{2} - \frac{3}{2} = 0 } \\ \\ \large \bold{2x {}^{2} - 3 = 0 } \\ \large \bold \pink{ \underline{ \underline{divide \: \: }}} \\ \\ \large \bold{ \frac{2x {}^{4} - 2x {}^{3} - 7x {}^{2} + 3x + 6 }{2x {}^{2} - 3 } } \\ \\ \large \bold \green{quotient = \: \: {x}^{2} - x - 2 = 0 } \\ \\ \large \bold \green{remainder = 0} \\ \\ \large \bold \blue{factorise \: \: quotient \: \: into \: \: middle \: \: term \: \:split } \\ \\ \large \bold{ {x}^{2} - x - 2 = 0 } \\ \large \bold{ {x}^{2} -2 x + x - 2 = 0 } \\ \large \bold{x(x -2) + 1(x - 2) = 0} \\ \large \bold{(x + 1)(x - 2) = 0} \\ \large \bold{x = - 1 \: \: and \: \: x = 2} \\ \\ \large \bold \green{ \underline{ \underline{all \: \: zeroes \: \: are \: \: - \sqrt { \frac{3}{2} } \: \: and \: \: \sqrt{ \frac{3}{2} } \: \: and \: \: 2 \: \: and \: \: - 1 }}}$