Answer
Zeros: $-2,1$
$x$-intercepts: $-2,1$
Work Step by Step
To find the zeros of a function $f$, solve the equation $f(x)=0$
The zeros of the function are also the $x-$intercepts.
Let $g(x)=0$:
$$x^6+7x^3-8=0$$
Let $u=x^3$, the original equation becomes
$$u^2+7u-8=0$$
By factoring
$$(u+8)(u-1) = 0$$
Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
\begin{align*}
u+8 &=0 &\text{ or }& &u-1=0\\
u &= -8 &\text{ or }& &u=1\\
\end{align*}
To solve for $x$, we use $u=x^3$
$$\because u = x^3$$
$$\therefore x = u^{\frac{1}{3}}$$
For $u=-8$
$$x=(-8)^{\frac{1}{3}} $$
$$x= -2$$
For $u=1$
$$x=(1)^{\frac{1}{3}}$$
$$x= 1$$
$\therefore x = -2,1$
