If the remainder on division of $x^{3} + 2 x^{2} + k x + 3$ by $x - 3$ is $21,$ find the quotient and value $8 k$

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Solution

Let $Q (x)$ be the quotient
According to the dividend rule
$x^{3} + 2 x^{2} + k x + 3 = Q (x) (x - 3) + 21$
Put $x = 3, 3^{3} + 2 (3)^{2} + k (3) + 3 = 0 + 21 ⟹ k = - 9 ⟹ 8 k = - 72$
$Q (x) = \frac{x^{3} + 2 x^{2} - 9 x - 18}{x - 3} = x^{2} + 5 x + 6$

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