Determine whether $(x + 1)$ is a factor of the polynomial:
$3 x^{3} + 8 x^{2} - 6 x - 5$

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Solution

According to factor theorem, when $f (x)$ is divided by $(x + 1), f (- 1) = 0$
$f (x) = 3 x^{3} + 8 x^{2} - 6 x - 5$
$f (- 1) = 3 (- 1)^{3} + 8 (- 1)^{2} - 6 (- 1) - 5$
$= - 3 + 8 + 6 - 5 = 6$
$∵ f (- 1) \neq 0, (x + 1)$ is not a factor of the given polynomial.

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