If $3$ and $- 3$ are two roots of the polynomial $(x^{4} + x^{3} - 11 x^{2} - 9 x + 18)$ , find all the roots of the given polynomial.

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Solution

$e q^{n} = x^{4} + x^{3} - 11 x^{2} - 9 x + 18$
Now
$x^{3} (x - 3) + 4 x^{2} (x - 3) + x (x - 3) - 6 (x - 3)$
$(x^{3} + 4 x^{2} + x - 6) (x - 3)$
${x^{2} (x + 3) + x (x + 3) - 2 (x + 3)} (x - 3)$
$(x^{2} + x - 2) (x + 3) (x - 3)$
$(x + 2) (x - 1) (x + 3) (x - 3) = 0$
so roots are
$x = - 2, 1, 3, - 3$

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