If the factorized form of $x^{3} + 3 x^{2} - 9 x + 5$ is $(x - 1) (x^{2} + a x - 5)$ . Find $a$

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Solution

$x^{3} + 3 x^{2} - 9 x + 5 = (x + 1) (x^{2} + a x - 5)$
Comparing coefficient of each term on LHS and RHS,
$\Rightarrow x^{3} + 3 x^{2} - 9 x + 5 = x^{3} + a x^{2} - 5 x - x^{2} - a x + 5$
$= x^{3} + (a - 1) x^{2} - (5 + a) x + 5$
$\Rightarrow (a - 1) = 3$
$\Rightarrow a = 4.$
Hence, the answer is $4.$

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