Obtain all other zeroes of $f (x) = x^{4} - 3 x^{3} - x^{2} + 9 x - 6$ if two of its zeroes are $- \sqrt{3}$ and $\sqrt{3}$ .

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Solution

Since two zeroes are $- \sqrt{3}$ and $\sqrt{3}$ . So $(x + \sqrt{3}) (x - \sqrt{3}) = x^{2} - 3$ , is a factor of the given polynomial.
$x^{2} - 3 x + 2$
__________
$x^{2} - 3) x^{4} - 3 x^{3} - x^{2} + 9 x - 6$
$- x^{4}$ $- 3 x^{2}$
$+$
_
$- 3 x^{3} + 2 x^{2} + 9 x - 6$
$- 3 x^{3}$ $+ 9 x$
$+$ $-$

$2 x^{2} - 6$
$2 x^{2} - 6$
$-$ $+$
__________
$0$
Quotient $x^{2} - 3 x + 2 = x^{2} - 2 x - x + 2$
$= x (x - 2) - 1 (x - 2)$
$= (x - 2) (x - 1)$
$(x - 2) = 0$
or $(x - 1) = 0$
$x = 2$ or $x = 1$ .

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