Determine whether $3 x - 2$ a factor of $3 x^{3} + x^{2} - 20 x + 12$

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Solution

Let $f (x) = 3 x^{3} + x^{2} - 20 x + 12$ .
If $3 x - 2$ is a factor of $f (x)$ then $f (\frac{2}{3}) = 0$

$f (\frac{2}{3}) = 3 (\frac{2}{3})^{3} + (\frac{2}{3})^{2} - 20 (\frac{2}{3}) + 12 = \frac{8}{9} + \frac{4}{9} - \frac{40}{3} + 12 = \frac{12}{9} - \frac{40}{3} + 12$

$= \frac{4 - 40 + 36}{3} = 0$
Hence $3 x - 2$ is a factor of $3 x^{3} + x^{2} - 20 x + 12$ .

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