Identify perfect square trinomial I've read the definition perfect square for numbers, that is a number is a perfect square if it is the product of two equal integers. Now I'm studying perfect square trinomial so I'm confused about the definition of a perfect square when a literal is involved. I've seen examples like $x^2$ is a perfect square since x^2=(x)^2, but then every term involving a literal is a perfect square since for instance $x=(\sqrt{x})^2$ and $x$ can be any real number, now if this is not a perfect square then how do I dentify perfect square trinomial since in very book I've read they say the first term must be a perfect square but then $x+4\sqrt{x}+4=(\sqrt{x} + 2)^2$ is a perfect square trinomial. So since I think every term involving a literal can be a perfect square then how to identify a perfect square trinomial? Even the second term $2xy$ can be seen as a perfect square sinnce $2xy = (\sqrt{2xy} )^2$ so it makes no sense to tell to identify the perfect squares.

Great question. Here's the subtlety - a binomial is a _polynomial_ with two terms. $\sqrt{x}$ is not allowed in polynomials, because $\sqrt{x} = x^{1/2}$, and fractional exponents are not allowed in polynomials. So while it is true that $x$ may take the value of something that makes $\sqrt{x}$ simplify nicely (i.e., $x$ may be a perfect square), that actually doesn't matter.

Similarly, a trinomial is a _polynomial_ with three terms, and a perfect square trinomial is a trinomial that is equal to the square of a binomial.

So $x^2 + 4x + 4$ is a perfect square trinomial because it's equal to $(x+2)^2$. But $x^2 + 4\sqrt{x} + 4$ is not even a trinomial, because of the $\sqrt{x}.$