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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 $6$.
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