What must be subtracted from 16x³ - 8x² + 4x + 7 so that the resulting expression has (2x + 1) as a factor?

Let the number to be subtracted from 16x³ - 8x² + 4x + 7 be a.

Resulting polynomial [f(x)] = 16x³ - 8x² + 4x + 7 - a

Given,

Factor: 2x + 1

⇒ 2x + 1 = 0

⇒ 2x = -1

⇒ x = $-\dfrac{1}{2}$.

Since, 2x + 1 is a factor.

Thus, on dividing 16x³ - 8x² + 4x + 7 - a by 2x + 1, remainder = 0.

$\therefore f\Big(\dfrac{-1}{2}\Big) = 0 \\[1em] \Rightarrow 16\Big(\dfrac{-1}{2}\Big)^3 - 8\Big(\dfrac{-1}{2}\Big)^2 + 4\Big(\dfrac{-1}{2}\Big) + 7 - a = 0 \\[1em] \Rightarrow 16\Big(\dfrac{-1}{8}\Big) - 8\Big(\dfrac{1}{4}\Big) - 2 + 7 - a = 0 \\[1em] \Rightarrow -2 - 2 - 2 + 7 - a = 0 \\[1em] \Rightarrow -6 + 7 - a = 0 \\[1em] \Rightarrow a = 1.$

Hence, the required number to be subtracted from the polynomial = 1.