Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:

Solution:

Here, $f(x)=3 x^{3}+x^{2}-20 x+12$

g(x) = 3x - 2

To prove that g(x) is the factor of f(x),

we should show ⟹ f(2/3) = 0

here, 3x - 2 = 0

⟹ 3x = 2

⟹ x = 2/3

Substitute the value of x in f(x)

$f(2 / 3)=3(2 / 3)^{3}+(2 / 3)^{2}-20(2 / 3)+12$

= 3(8/27) + 4/9 − 40/3 + 12

= 8/9 + 4/9 − 40/3 + 12

= 12/9 − 40/3 + 12

Taking L.C.M

$=\frac{12-120+108}{9}$

$=\frac{120-120}{9}$

= 0

Since, the result is 0 g(x) is the factor of f(x)