Question

What must be subtracted from $$16x^{3} - 8x^{2} + 4x + 7$$ so that the resulting expression has $$(2x + 1)$$ as a factor?

Answer 1

Let the amount subtracted be $$k$$ and let $$f(x) = 16x^{3} - 8x^{2} + 4x + 7$$.

Since $$2x + 1 = 0$$ gives $$x = -\frac{1}{2}$$, the value at this point must be zero.

$$f\left(-\frac{1}{2}\right) = 16\left(-\frac{1}{2}\right)^3 - 8\left(-\frac{1}{2}\right)^2 + 4\left(-\frac{1}{2}\right) + 7$$

$$= -2 - 2 - 2 + 7$$

$$= 1$$

Thus, $$1 - k = 0$$, so $$k = 1$$.

Indeed, $$16x^{3} - 8x^{2} + 4x + 7 - 1 = (2x + 1)(8x^{2} - 8x + 6).$$