Question:
If $\sqrt{5}$ and $-\sqrt{5}$ are two zeroes of the polynomial $x^{3}+3 x^{2}-5 x-15$, then its third zero is
(a) 3
(b) −3
(c) 5
(d) −5
Solution:
Let $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$ be the given zeros and $\gamma$ be the third zero of the polynomial $x^{3}+3 x^{2}-5 x-15$. Then,
By using $\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$
$\alpha+\beta+\gamma=\frac{-3}{1}$
$\alpha+\beta+\gamma=-3$
Substituting $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$ in $\alpha+\beta+\gamma=-3$
We get
$\sqrt{5}-\sqrt{5}+\gamma=-3$
$\gamma=-3$
Hence, the correct choice is $(b)$
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