Question

Find all the zeroes of the polynomial 3y4 +6y3 –2y2–10y–5, if two of its zeroes are √5/3 and-√5/3

Answer 1

Step-by-step explanation:

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Answer 2

$Given \: polynomial \:is$

$p(x) = 3x^{4}+6x^{3}-2x^{2}-10x-5 \: --(1)$

The degree of the given polynomial is 4, so it has at most 4 zeroes .

$The \: two \:zeroes \:of \: given \: polynomial$

$are \: \sqrt{\frac{5}{3}} \: and \: -\sqrt{\frac{5}{3}}$

$The \: equation \:of \:the \: polynomial$ $whose\: zeroes \:are \:\sqrt{\frac{5}{3}} \: and$

$-\sqrt{\frac{5}{3}}\:is$

$\Big( x + \sqrt{\frac{5}{3}}\Big) \Big( x - \sqrt{\frac{5}{3}}\Big)$

$= x^{2} - \frac{5}{3}\: --(2)$

Divide (1) with (2) , we get ,

[ See the attachment ]

$p(x) = \Big( x^{2} - \frac{5}{3}\Big) (3x^{2}+6x+3)$

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

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

$= 3 \Big( x + \sqrt{\frac{5}{3}}\Big) \Big( x - \sqrt{\frac{5}{3}}\Big) (x+1)(x+1)$

$\therefore The \:zeroes \:of \:the \: given \:$

$polynomial \:are \: \sqrt{\frac{5}{3}},- \sqrt{\frac{5}{3}}, -1 \:and \:-1$

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