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

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Solution

Let the other roots be $a, b$
Sum of roots = $a + b + \sqrt{\frac{5}{3}} - \sqrt{\frac{5}{3}} = a + b = - \frac{- 6}{3} = \frac{- 6}{3}$

Product of roots = $a b = \sqrt{\frac{5}{3}} * - \sqrt{\frac{5}{3}} = \frac{- 5}{3} \Rightarrow a b = 1 \Rightarrow b = \frac{1}{a}$

$a + \frac{1}{a} = \frac{- 6}{3} \Rightarrow \frac{a^{2} + 1}{a} = \frac{- 6}{3} \Rightarrow a = - 1$

Hence, $b = \frac{1}{a} = - 1$

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Obtain all other zeroes of 3x4+6x3−2x2−10x−5, if two of its zeroes are √53 and −√53

Let the other roots be a,bSum of roots = a+b+√53−√53=a+b=−−63=−63Product of roots = ab=√53∗−√53=−53⇒ab=1⇒b=1aa+1a=−63⇒a2+1a=−63⇒a=−1Hence, b=1a=−1

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