What number must be subtracted from 2x² - 5x so that resulting polynomial leaves remainder 2 when divided by 2x + 1 ?

Let the number to be subtracted be a.

So, polynomial = 2x² - 5x - a

By remainder theorem, on dividing f(x) by (x - b) , remainder = f(b)

∴ On dividing, f(x) = 2x² - 5x - a by (2x + 1) or 2(x - $\big(-\dfrac{1}{2}\big)$)

Remainder = f$\big(-\dfrac{1}{2}\big)$

Given, remainder = 2

$\therefore 2\big(-\dfrac{1}{2}\big)^2 - 5\big(-\dfrac{1}{2}\big) - a = 2 \\[1em] \Rightarrow 2\big(\dfrac{1}{4}\big) + \dfrac{5}{2} - a = 2 \\[1em] \Rightarrow \dfrac{1}{2} + \dfrac{5}{2} - a = 2 \\[1em] \Rightarrow \dfrac{6}{2} - a = 2 \\[1em] \Rightarrow 3 - a = 2 \\[1em] \Rightarrow a = 3 - 2 \\[1em] a = 1.$

Hence, the value of a is 1.