Examine whether $x + 2$ is a factor of $x^{3} + 2 x^{2} + 3 x + 6$

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Solution

By factor theorem,if the value of polynomial $p (x)$ is zero at $x = c$ , then $g (x) = x - c$ is a factor of $p (x)$ .
Given $p (x) = x^{3} + 2 x^{2} + 3 x + 6$ and $g (x) = x + 2$
To find value of $c$ ,
$g (x) = 0$ then $c = - 2$
if we replace $x$ by $- 2$ we get,
$p (- 2) = (- 2)^{3} + 2 (- 2)^{2} + 3 (- 2) + 6$
$p (- 2) = - 8 + 8 - 6 + 6$
$p (- 2) = 0$
Therefore, $x + 2$ is factor of $x^{3} + 2 x^{2} + 3 x + 6$

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Examine whether x+2 is a factor of x3+2x2+3x+6

Examine whether $x + 2$ is a factor of $x^{3} + 2 x^{2} + 3 x + 6$