Answer
Please see "step by step"
Work Step by Step
By The Linear Factorization Theorem
If $f(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$, then
$f(x)=a_{n}(x-c_{1})(x-c_{2})(x-c_{3})$
where $c_{1}, c_{2}, c_{3}$ are complex numbers (possibly real and not necessarily distinct).
Imaginary roots, if they exist, occur in conjugate pairs.
(If $a+bi$ is a root then $a-bi$ is also a root).
So, there are 3 roots, and they can't all be complex.
1. If none are complex, there are 3 real roots. (there is a real root)
2. If one is a complex root, then there must be another complex root, its conjugate,
The third root can not be complex because it has no pair...(a conjugate would mean there are 4 roots, which can not be for degree n=3)
...so it must be real (no imaginary component).
(there is a real root)
In any case, for n=3, there is a real root.
