Answer
$\dfrac{2}{3}$ is not a zero of the function by the factor theorem.
Work Step by Step
The factor theorem states that when $f(a)=0$, then we have $(x-a)$ as a factor of $f(x)$ and when $(x-a)$ is a factor of $f(x)$, then $f(a)=0$.
We are given the function $f(x)=x^7+6x^5-x^4+x+2$
We simplify the given equation as follows:
$f(\dfrac{2}{3})=(\dfrac{2}{3})^7+6(\dfrac{2}{3})^5-(\dfrac{2}{3})^4+\dfrac{2}{3}+2 \\= 3.318$
This implies that $f(\dfrac{2}{3}) \ne 0$
So, $\dfrac{2}{3}$ is not a zero of the function by the factor theorem.
