Obtain all other zeros of $(x^{3} + 4 x^{3} - 2 x^{2} - 20 x - 15)$ if two of its zeros are $\sqrt{5} a n d - \sqrt{5}$ .

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Solution

$\sqrt{5} a n d - \sqrt{5}$ are the zeros of the polynomial $(x^{3} + 4 x^{3} - 2 x^{2} - 20 x - 15)$

Therefore, $(x - \sqrt{5}) (x + \sqrt{5}) = x^{2} - 5$ will divide the given polynomial completely.

Dividing $(x^{3} + 4 x^{3} - 2 x^{2} - 20 x - 15)$ by $x^{2} - 5$ , we get

Quotient $= x^{2} + 4 x + 3 = x^{2} + 3 x + x + 3$

$= x (x + 3) + (x + 3) = (x + 3) (x + 1)$

Other zeros of the given polynomial are the zeros of q(x)

Therefore, $x = - 3, - 1$

Thus, the zeros of the given polynomial are $\sqrt{5}, - \sqrt{5}, - 3, - 1$

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Obtain all other zeros of (x3+4x3−2x2−20x−15) if two of its zeros are √5 and −√5.

Obtain all other zeros of $(x^{3} + 4 x^{3} - 2 x^{2} - 20 x - 15)$ if two of its zeros are $\sqrt{5} a n d - \sqrt{5}$ .