What number must be added to 2x³ - 3x² - 8x so that resulting polynomial leaves the remainder 10 when divided by 2x + 1?

Given,

⇒ 2x + 1 = 0

⇒ 2x = -1

⇒ x = $-\dfrac{1}{2}$

Let number added be a.

Polynomial = 2x³ - 3x² - 8x + a

By remainder theorem,

When a polynomial p(x) is divided by (x - a), then the remainder = f(a).

$\therefore 2.\Big(-\dfrac{1}{2}\Big)^3 - 3.\Big(-\dfrac{1}{2}\Big)^2 - 8.\Big(-\dfrac{1}{2}\Big) + a = 10\\[1em] \Rightarrow 2.\Big(-\dfrac{1}{8}\Big) - 3.\Big(\dfrac{1}{4}\Big) + \Big(\dfrac{8}{2}\Big) + a = 10\\[1em] \Rightarrow -\dfrac{2}{8} - \Big(\dfrac{3}{4}\Big) + 4 + a = 10\\[1em] \Rightarrow -\dfrac{1}{4} - \Big(\dfrac{3}{4}\Big) + 4 + a = 10\\[1em] \Rightarrow \dfrac{-1 - 3}{4} + 4 + a = 10\\[1em] \Rightarrow \dfrac{-4}{4} + 4 + a = 10\\[1em] \Rightarrow -1 + 4 + a = 10\\[1em] \Rightarrow 3 + a = 10\\[1em] \Rightarrow a = 10 - 3\\[1em] \Rightarrow a = 7$

Hence, the number to be added = 7.