Using factor theorem, show that:

(3 - 2x) is a factor of (2x³ - 9x² + x + 12).

Let f(x) = (2x³ - 9x² + x + 12)

Given,

Divisor:

⇒ 3 - 2x = 0

⇒ 2x = 3

⇒ x = $\dfrac{3}{2}$

By factor theorem,

(x - a) is a factor of f(x), if f(a) = 0.

Substituting x = $\Big(\dfrac{3}{2}\Big)$ in f(x), we get :

$\Rightarrow f\Big(\dfrac{3}{2}\Big) = 2\Big(\dfrac{3}{2}\Big)^3 - 9\Big(\dfrac{3}{2}\Big)^2 + \Big(\dfrac{3}{2}\Big) + 12 \\[1em] = 2\Big(\dfrac{27}{8}\Big) - 9\Big(\dfrac{9}{4}\Big) + \Big(\dfrac{3}{2}\Big) + 12 \\[1em] = \dfrac{27}{4} - \dfrac{81}{4} + \dfrac{3}{2} + 12 \\[1em] = \Big(\dfrac{27 - 81 + 6 + 48}{4}\Big) \\[1em] = \dfrac{81 - 81}{27} \\[1em] = 0.$

Since, f$\Big(\dfrac{3}{2}\Big)$ = 0, thus (3 - 2x) is a factor of f(x).

Hence, proved that (3 - 2x) is factor of 2x³ - 9x² + x + 12.