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

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

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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