Question

For what value of $$a$$ is the polynomial $$(2x^{3} + ax^{2} + 11x + a + 3)$$ exactly divisible by $$(2x - 1)$$?

Answer 1

Let $$f(x) = 2x^{3} + ax^{2} + 11x + a + 3$$.

The factor $$2x - 1 = 0$$ gives $$x = \frac{1}{2}$$.

Therefore, by the Factor Theorem,

$$f\left(\frac{1}{2}\right) = 0$$

$$2\left(\frac{1}{2}\right)^3 + a\left(\frac{1}{2}\right)^2 + 11\left(\frac{1}{2}\right) + a + 3 = 0$$

$$\frac{1}{4} + \frac{a}{4} + \frac{11}{2} + a + 3 = 0$$

$$\frac{5a+35}{4} = 0$$

$$\Rightarrow 5a + 35 = 0$$

$$\Rightarrow a = -7$$