Math, asked by supergirl1234, 9 months ago

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

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Answered by parijindal47
5

Step-by-step explanation:

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Answered by mysticd
1

 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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