Obtain all zeroes of $3 x^{4} - 15 x^{3} + 13 x^{2} + 25 x - 30$ if two of its zeroes are $\sqrt{\frac{5}{3}}$ & $- \sqrt{\frac{5}{3}}$

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Solution

By factor theorem,if $\sqrt{\frac{5}{3}}$ and $- \sqrt{\frac{5}{3}}$ are the zeroes of the polynomial then $x - \sqrt{\frac{5}{3}} = 0$ and $x + \sqrt{\frac{5}{3}} = 0$
$\Rightarrow (x - \sqrt{\frac{5}{3}}) (x + \sqrt{\frac{5}{3}}) = 0$ is one of the factor of the polynomial.
$\Rightarrow x^{2} - \frac{5}{3} = 0$
The other factor are calculated by dividing the polynomial by $x^{2} - \frac{5}{3}$
$3 x^{4} - 15 x^{3} + 13 x^{2} + 25 x - 30 \div x^{2} - \frac{5}{3}$
The other factor is $3 x^{2} - 15 x - 18$
$3 (x^{2} - 5 x - 6)$
$= 3 (x^{2} - 6 x + x - 6)$
$= 3 (x - 6) (x + 1)$
$∴$ the factors of the polynomial are $(x - \sqrt{\frac{5}{3}}), (x + \sqrt{\frac{5}{3}}), (x - 6), (x + 1)$
The other zeroes of the polynomial are $6, - 1$

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Obtain all zeroes of 3x4−15x3+13x2+25x−30 if two of its zeroes are √53 & −√53

Obtain all zeroes of $3 x^{4} - 15 x^{3} + 13 x^{2} + 25 x - 30$ if two of its zeroes are $\sqrt{\frac{5}{3}}$ & $- \sqrt{\frac{5}{3}}$