When is a real number algebraic? By definition - A real number is algebraic if it is a root of a non-zero polynomial equation with rational coefficients. What does non-zero polynomial equation mean?
Well, an equation f(x) = x -5, becomes zero when x = 5, so this is a zero polynomial equation. Is the definition saying that the equation should not equal zero in any case?
Can someone clarify this?

A zero polynomial is a polynomial which gives $0$ for all values of $x$,
Basically $P(x)=0$ means that no matter what $x$ you put in there, the result will always be zero. For your example,
$P(x)=x-5$ is not a zero polynomial because there is only a finite number of $x$'s for which the polynomial will be $0$. In this case there is only one such $x$. And that is $5$.
And a real number is algebraic if there exists a non-$\color{green}{\textrm{constant}}$ polynomial with rational coefficients, such that the the real number is one of the roots of the polynomial.
