When $p (x) = x^{3} + {a x}^{2} + 2 x + a$ is divided by $(x + a),$ the remainder is _________.

0

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a

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- a

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2 a

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Solution

The correct option is D $- a$
Given: $p (x) = x^{3} + a x^{2} + 2 x + a$

By remainder theorem, when a polynomial, $p (x)$ is divided by a linear polynomial $(x + a)$ , then the remainder of that division will be equivalent to $p (a)$ .

Here, $p (x)$ is divided by a linear polynomial $(x + a)$ , then the remainder of the division will be equivalent to $p (- a)$ .

Then, remainder
$p (- a)$ $= {(- a)}^{3} + a {(- a)}^{2} + 2 (- a) + a$
$= - a^{3} + a^{3} - 2 a + a = - a$ .

Therefore, option $C$ is correct.

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