What number should be subtracted from 2x³ - 5x² + 5x so that the resulting polynomial has 2x - 3 as a factor?

Let the number to be subtracted be a.

f(x) = 2x³ - 5x² + 5x - a

If, (2x - 3) or 2(x - ($\dfrac{3}{2}$)) is a factor of f(x) then f($\dfrac{3}{2}$) = 0, by factor theorem

$\therefore 2\big(\dfrac{3}{2}\big)^3 - 5\big(\dfrac{3}{2}\big)^2 + 5(\dfrac{3}{2}) - a = 0 \\[1em] \Rightarrow 2\big(\dfrac{27}{8}\big) - 5\big(\dfrac{9}{4}\big) + \dfrac{15}{2} - a = 0 \\[1em] \Rightarrow \dfrac{27}{4} - \dfrac{45}{4} + \dfrac{15}{2} - a = 0 \\[1em] \Rightarrow \dfrac{27 - 45 + 30 - 4a}{4} = 0 \\[1em] \Rightarrow 12 - 4a = 0 \\[1em] \Rightarrow 4a = 12 \\[1em] a = 3.$

Hence, the number to be subtracted is 3.