Find the value of $m$ in order that $x^{4} - 2 x^{3} + 3 x^{2} - m x + 5$ may be exactly divisible by $x - 3$ .

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Solution

Let , $p (x) = x^{4} - 2 x^{3} + 3 x^{2} - m x + 5$
If $(x - 3)$ exactly divides $p (x),$ then for $x = 3$ , $p (x) = 0$
i.e. $p (3) = 0$
$3^{4} - 2 \times 3^{3} + 3 \times 3^{2} - 3 m + 5 = 0$
$81 - 54 + 27 - 3 m + 5 = 0$
$- 3 m + 59 = 0$
$m = \frac{59}{3}$

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