In the wikipedia, they define 'Algebraic equation' like this:
In mathematics, an algebraic equation or polynomial equation is an equation of the form
$P=0$
where P is a polynomial with coefficients in some field, often the field of the rational numbers.
is The form $P=0$ means the Algebraic equation's right side needs to be zero? or is it only notation for simplicity?
The reason that an algebraic equation is defined as $P(x) = 0$ where $P(x)$ is a polynomial has nothing to do with zero. Many polynomials $P(x)$ are defined as the product of other polynomials (eg. $P(x) = u(x) * v(x) * w(x)$), and since most standard number systems are defined without zero factors, that means that you can wash away a lot of complexity by finding where $u, v$, or $w$ are equal to zero.
$P(x) = 0$ is not special, as $P(x) = 1$ can be reduced to $P(x) - 1 = 0$ which is another polynomial. Or, even when two polynomials are matched (eg. $P(x) = Q(x)$), they can still be reduced to a single polynomial ($P(x) - Q(x) = 0$). The definition is merely that an algebraic equation is merely an equation constructed of a summation of terms defined as $cx^z$
where $c$ is a real number and $z$ is an integer. Which side the terms appear on is unimportant.