Find all the zeroes of the polynomial $3 x^{4} + 6 x^{3} - 2 x^{2} - 10 x - 5$ , if two of its zeroes are $\sqrt{\frac{5}{3}}$ $- \sqrt{\frac{5}{3}}$ .

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Solution

$P (x) = 3 x^{4} + 6 x^{3} - 2 x^{2} - 10 x - 5$
as we have given that two of its zeros are $\sqrt{5} / 3$ and $- \sqrt{5} / 3$ , so their factors are $(x - \sqrt{5} / 3)$ and $(x + \sqrt{5} / 3)$
The product of the factors;
$(x - \sqrt{5} / 3) . (x + \sqrt{5} / 3) = x^{2} - 5 / 3$
$= 3 x^{2} - 5 / 3 = \frac{1}{3} (3 x^{2} - 5)$
Now $g (x) = 3 x^{2} - 5$
dividing p(x) by g(x) we get
$q (x) = x^{2} + 2 x + 1$
by division algorithm
$p (x) = g (x) + q (x) + r (x)$
$= (3 x^{2} - 5) (x^{2} + 2 x + 1)$
$= (3 x^{2} - 5) (x^{2} + x + x + 1)$
$= (3 x^{2} - 5) (x (x + 1) + 1 (x + 1))$
$= (3 x^{2} - 5) (x + 1) (x + 1)$
$∴$ The other two zeros are -1 and -1

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Similar questions

Q. Find all zeroes of the polynomial

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if two of its zeroes are

\frac{\sqrt{5}}{3}

and

- \frac{\sqrt{5}}{3}

Q. Obtain all other zeroes of

3 x^{4} + 6 x^{3} - 2 x^{2} - 10 x - 5

, if two of its zeroes are

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Q. Obtain all the zeroes of

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, if of its zeroes are

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- \sqrt{\frac{5}{3}}

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

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

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Find all the zeroes of the polynomial $3 x^{4} + 6 x^{3} - 2 x^{2} - 10 x - 5$ , if two of its zeroes are $\sqrt{\frac{5}{3}}$ $- \sqrt{\frac{5}{3}}$ .