Find the factorss of $x^{4} + 3 x^{3} - 7 x^{2} - 27 x - 18$

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Solution

If $x = a$ is a zero, then $x - a$ is a factor
$f (x) = x^{4} + 3 x^{3} - 7 x^{2} - 27 x - 18$
$f (- 1) = 1 - 3 - 7 + 27 - 18 = 0$
$(x + 1)$ is a factor of $f (x)$
The other factors are calculated by using synthetic division.
$(x + 1) (x^{3} + 2 x^{2} - 9 x - 18)$
$= (x + 1) (x^{2} (x + 2) - 9 (x + 2))$
$= (x + 1) (x + 2) (x^{2} - 9)$
$= (x + 1) (x + 2) (x - 3) (x + 3)$

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