What number should be subtracted from 2x³ - 5x² + 5x + 8 so that the resulting polynomial has a factor 2x - 3 ?

Let no. to be subtracted be a.

So, resulting polynomial is 2x³ - 5x² + 5x + 8 - a.

By factor theorem,

A polynomial f(x) has a factor (x - a), if and only if, f(a) = 0.

Given,

Factor : 2x - 3 = 0

x = $\dfrac{3}{2}$

Substituting value of x = $\dfrac{3}{2}$ in 2x³ - 5x² + 5x + 8 - a, will give remainder = 0.

$\Rightarrow 2 \times \Big(\dfrac{3}{2}\Big)^3 - 5 \times \Big(\dfrac{3}{2}\Big)^2 + 5 \times \dfrac{3}{2} + 8 - a = 0 \\[1em] \Rightarrow 2 \times \dfrac{27}{8} - 5 \times \dfrac{9}{4} + \dfrac{15}{2} + 8 - a = 0 \\[1em] \Rightarrow \dfrac{27}{4} - \dfrac{45}{4} + \dfrac{15 + 16}{2} - a = 0 \\[1em] \Rightarrow -\dfrac{18}{4} + \dfrac{31}{2} - a = 0 \\[1em] \Rightarrow \dfrac{-18 + 62}{4} - a = 0 \\[1em] \Rightarrow \dfrac{44}{4} - a = 0 \\[1em] \Rightarrow 11 - a = 0 \\[1em] \Rightarrow a = 11.$

Hence, number to be subtracted is 11.