The polynomial $x^{3} + m x^{2} - x + 6$ has $(x - 2)$ as a factor and leaves a remainder $n$ when divided by $(x - 3)$ . Find the values of $m$ and $n$ .

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Solution

Let $p (x) = x^{3} + m x^{2} - x + 6$ as $(x - 2)$ is a factor of $p (x)$
$\Rightarrow p (2) = 0 \Rightarrow 8 + 4 m - 2 + 6 = 0 \Rightarrow 4 m = - 12 \Rightarrow m = - 3$

$x - 3) x^{3} - 3 x^{2} - x + 6 (x^{2} - 1$
$x^{3} - 3 x^{2}$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - x + 6$
$+ x + 3$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 3 = n$

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