Classify polynomials by number of terms

Polynomial expressions are classified by counting their nonzero terms after like terms have been combined: one term is a monomial, two terms a binomial, and three terms a trinomial; expressions with four or more terms may be described as polynomials with that number of terms. Terms are separated by addition or subtraction, so a negative term remains one term, and a constant is itself a term. The classification applies to standard polynomial expressions with nonnegative integer exponents, not rational or negative exponents.

Detailed Explanation: Classify polynomials by number of terms

To classify a polynomial, first combine like terms. Then count the nonzero terms.

1 term: monomial
2 terms: binomial
3 terms: trinomial
4 or more terms: polynomial with that number of terms

Terms are separated by addition or subtraction. A negative term still counts as one term.

Example

Classify:

3x^2+4x-2+5x^2-x

Step 1: Combine like terms.

Combine the $x^2$ terms:

3x^2+5x^2=8x^2

Combine the $x$ terms:

4x-x=3x

The constant $-2$ has no like term, so it stays as it is.

Thus, the expression becomes:

8x^2+3x-2

Step 2: Count the terms.

The terms are:

8x^2,\quad 3x,\quad -2

There are $3$ nonzero terms, so the expression is a trinomial.

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