It is given that $f(x)=x^3+3q{x}^2+2$, where $q$ is a real constant.

The equation $f(x)=0$ has $3$ distinct real roots.

Which of the following statements is/are necessarily true?

I The equation $f(x)+1=0$ has $3$ distinct real roots.

II The equation $f(x+1)=0$ has $3$ distinct real roots.

III The equation $f(-x)-1=0$ has $3$ distinct real roots.