What is transcendental equation/function?

Question

I looked up several sources on the internet.

A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have closed-form solutions.

And then transcendental function

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.

Unfortunately, I don't quite understand what it means "to satisfy polynomial equation"

I also checked several questions asked on this site. One of the most relevant is:

In simple English, what does it mean to be transcendental?

However, although OP asked about "transcendental function in layman terms" , the most pertinent answers mostly answer the question "What is transcendental number"

So I would like to ask you, if we use the most basic language possible, what is transcendental equation/function? And how do I determine whether one is a transcendental function/equation?

Answer 1

a transcendental function f(x) gives transcendental results for most rational x example: e^x, sin(x) etc. the simple seaming equation e^x=x or cos(x)=x have no formula for x as result, but must be calculated numerically. also you cn not rewrite e^x as a polynomial or a fraction of polynoms trula

Answer 2

A polynomial is an expression obtained by combining constants and variables by means of a finite number of additions and multiplications.

E.g. $3xy^3+2x-1$.

An algebraic function of a single variable $x$ is such that the relation to the dependent variable $y(x)$ can be expressed by a bivariate polynomial equation with integer coefficients.

E.g. $3x(y(x))^3+2x-1=0$, which can also be written $y(x)=\sqrt[3]{\dfrac{1-2x}{3x}}$.

In particular, the ratio of two polynomials in $x$ is an algebraic function, as is any expression involving only the four operations and radicals.

A transcendental function is one that is not algebraic.

E.g. $\sin(x)$ is transcendental because there is no polynomial $P$ such that $P(x,\sin(x))=0$.