Question

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

Answer 1

It is given that $x=-\frac{1}{2}$ is a zero of the polynomial $p\left(x\right)=8x^3-a{x}^2-x+2$ which means that $p(-\frac{1}{2})=0$.

Let us substitute $x=-\frac{1}{2}$ in the polynomial $p\left(x\right)=8x^3-a{x}^2-x+2$ as shown below:

$p(-\frac{1}{2})=8{(-\frac{1}{2})}^3-a{(-\frac{1}{2})}^2-(-\frac{1}{2})+2\Rightarrow 0=(8\times -\frac{1}{8})-(a\times \frac{1}{4})+\frac{1}{2}+2\Rightarrow 0=-1-\frac{a}{4}+\frac{1}{2}+2\Rightarrow 0=\frac{-4-a+2+8}{4}$

$\Rightarrow 0=\frac{6-a}{4}\Rightarrow 6-a=0\Rightarrow a=6$

Hence, $a=6$.