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

Given,

⇒ 2x + 1 = 0

⇒ x = -$\dfrac{1}{2}$.

Let number added be a.

Polynomial = 2x³ - 3x² - 8x + 3 + a

By remainder theorem,

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

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

Hence, no. to be added = 4.