Determine the value of m if (x-1) is a factor $x^{3} + 5 x^{2} + m n + 4$ .

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Solution

Let $p (x) = x^{3} + 5 x^{2} + m x + 4$ . Since (x-1) is a factor of p(x), the remainder p(1)=0. Now,
$p (1) = 0 \to (1)^{3} + 5 (1)^{2} + m (1) + 4 = 0$
$\to 1 + 5 + m + 4 = 0$
$m + 10 = 0$
$∴ m = - 10$

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