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Obtain all other zeroes of 3x^4 + 6x^3 – 2x^2 – 10x – 5, if two of its zeroes are

Q3  Obtain all other zeroes of 3x^4 + 6x^3 - 2x^2 - 10x - 5, if two of its zeroes are \sqrt {\frac5{3}} \: \:and \: \: - \sqrt {\frac{5}{3}}

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S Sayak

Two of the zeroes of the given polynomial are \sqrt {\frac5{3}} \: \:and \: \: - \sqrt {\frac{5}{3}}.

Therefore two of the factors of the given polynomial are x-\sqrt{\frac{5}{3}} and x+\sqrt{\frac{5}{3}}

(x+\sqrt{\frac{5}{3}})\times (x-\sqrt{\frac{5}{3}})=x^{2}-\frac{5}{3}

x^{2}-\frac{5}{3}   is a factor of the given polynomial.

To find the other factors we divide the given polynomial with 3\times (x^{2}-\frac{5}{3})=3x^{2}-5

 

The quotient we have obtained after performing the division is x^{2}+2x+1

\\x^{2}+2x+1\\ =x^{2}+x+x+1\\ =x(x+1)+(x+1)\\ =(x+1)^{2}

(x+1)2 = 0

x = -1

The other two zeroes of the given polynomial are -1.

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