If $x = -\frac{1}{2}$ is a zero of the polynomial $p(x) = 8x^3 - ax^2 - x + 2$, find the value of $a$.

Given:

The given polynomial is $p(x) = 8x^3 - ax^2 - x + 2$.

$x = -\frac{1}{2}$ is a zero of the polynomial $p(x) = 8x^3 - ax^2 - x + 2$.

To do:

We have to find the value of $a$.

Solution:

The zero of the polynomial is defined as any real value of $x$, for which the value of the polynomial becomes zero.

Therefore,

Zero of the polynomial $p(-\frac{1}{2})=0$

$8(-\frac{1}{2})^{3}-a(-\frac{1}{2})^{2}-(-\frac{1}{2})+2=0$

$\Rightarrow 8 \times(-\frac{1}{8})-a \times \frac{1}{4}+\frac{1}{2}+2=0$

$\Rightarrow -1-\frac{a}{4}+\frac{1}{2}+2=0$

$\Rightarrow \frac{3}{2}-\frac{a}{4}=0$

$\Rightarrow \frac{a}{4}=\frac{3}{2}$

$\Rightarrow a=\frac{3 \times 4}{2}$

$\Rightarrow a=6$

The value of $a$ is $6$.