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$obtain \: all \: the \: zeroes \: of \: \\ 3x {}^{4} + 6x {}^{3} - 2x {}^{2} - 10x - 5 \\ if \: two \: of \: its \: zeroes \: are \: \sqrt{ \frac{5}{3} } and \\ - \sqrt{ \frac{5}{3} }.$
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Answer 1

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The other zeros are - 1 and - 1

Refer to the attachment

Also refer to the part marked important

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Hope this helps ✌️

Answer 2

Hello dear ,

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$it \: is \: given \: that \: \sqrt{ \frac{5}{3} } \: and \: - \sqrt{ \frac{5}{3} } \: are \: two \: zeros \: of \: f(x). \\ therefore. \: (x - \sqrt{ \frac{5}{3} } )(x + \sqrt{ \frac{5}{3} } ) \: isa \: factor \: of \: f(x). \\ \: but \: . \: (x - \sqrt{ \frac{5}{3} } )(x + \sqrt{ \frac{5}{3} } ) = ( {x}^{2} - \frac{5}{3} ) = \frac{1}{3} (3 {x}^{2} - 5)$


$therefore. \: 3 {x}^{2} - 5 \: is \: a \: factor \: of \: f(x). \: \\ \\ let \: us \: now \: divide \: f(x) \: by \: 3 {x}^{2} - 5. \\ \\ usig \: long \: division \: method \: \: we \: obtain \:$


( See the attachment ) ●●●●●


Clearly, Quotient = x^2 +2x +1 and Remainder = 0 .

By division algorithm, we obtain

$f(x) = (3 {x}^{2} - 5)( {x}^{2} + 2x + 1) + 0 \\ \\ = > f(x) = ( \sqrt{3} x + \sqrt{5} )( \sqrt{3} x - \sqrt{5} )( {x + 1)}^{2}$


Thus,

$the \: factors \: of \: f(x) \: are \: \sqrt{3} x + \sqrt{5} \: . \: \sqrt{3} x - \sqrt{5} \: . \: x + 1 \: and \: x + 1$


Equating each factor to zero, we obtain

$x = - \sqrt{ \frac{5}{3} } \: \: . \: \: \sqrt{ \frac{5}{3} } \: \: . \: \: - 1 \: \: . \: \: - 1.$


$hence \: the \: zeros \: of(x) \: are \: - \sqrt{ \frac{5}{3} } \: \: . \: \: \sqrt{ \frac{5}{3} } \: \: . \: \: - 1 \: \: . \: \: - 1.$


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Hope it helps u !!!

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have a wonderful day ahead

# Nikky