When we classify a polynomial as binomial, trinomial, etc., do we have to simplify it first?

Question

When we classify a polynomial as binomial, trinomial, etc., do we have to simplify it first?

Example: $3x$ is a monomial. Then $2x + x$ is a binomial? Then $x + x + x$ a trinomial?

I have not read anywhere where they say you must simplify first.

If the above is correct then $1+1+1$ would be a trinomial?

Answer 1

Is $3+4$ a prime? Is $11+43$ even?

The two polynomials $3x$ and $x+x+x$ are not just related through a series of certain operations. They are the exact same polynomial. So any property that cares about the actual polynomial, rather than typographical details, will stay true for both of them simultaneously or neither of them simultaneously.

Some properties do care about typography. They usually specify something like "written as" or "in X form". Their value lies mostly in convenience and in shaping the mentality of the writer / reader (making certain operations easier to perform, making it easier to confirm certain properties hold, suggesting the next steps to take, etc.), rather than actual objective math. For instance, "written as a square", where $x^2 + 2x + 1$ is not written as a square and $(x+1)^2$ is. But crucially they are both squares, because they are the same polynomial.

So $x+x+x$ is a monomial, but it's not written as a monomial.

Answer 2

For monomials, binomials, trinomials you don't just count the terms (like 3x + 4x + 5x + 6x + 7x = 5 terms). The first thing you look at is the highest degree of a power in these terms - in your case they are all degree 1, like 3x^1, so it is at worst monomial. On the other hand, the single term x^3 makes it a trinomial already.

After that, if you have detected it is for example a trinomial in the worst case because you had third powers but no fourth or higher powers, you add up all the constant factors with that power. Then for example 3x^3 + 2x - x^3 - 2x^3 is at worst a trinomial, but the constant factors 3, -1 and -2 with the third powers add up to 0, so you check everything again but ignoring the cubic powers. In this case it is a monomial.

BTW Trinomials usually have four terms, like a + bx + cx^2 + dx^3. Not three. The "tri" = "three" refers to the highest power that was used, not the number of terms.