What is the remainder when the polynomial $x^{3} - 2 x^{2} + x - 4$ is divided by $(x - 2)$ ? What number is to be added to the given polynomial so that $(x - 2)$ is a factor of it?

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Solution

$p (x) = x^{3} - 2 x^{2} + x - 4$
$p (2) = {(2)}^{3} - 2 {(2)}^{2} + 2 - 4 = 8 - 8 + 2 = - 2$
Thus, when the polynomial $x^{3} - 2 x^{2} + x - 4$ is divided by $(x - 2)$ , the remainder is $- 2$ . And hence, $2$ should be added to the given polynomial so that $(x - 2)$ is a factor of it.

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