Math101 · Independent Ontario learning library
Math101Binomial
A binomial is a polynomial with exactly two nonzero unlike terms after simplification. Examples include $x+5$, $3a^2-2a$, and $p^3q+7$.
Recognizing binomials selects useful multiplication, factoring, and expansion patterns. Term count and degree are independent structural descriptions.
Intuition and core definition
A binomial is a polynomial with exactly two nonzero unlike terms after simplification. Examples include $x+5$, $3a^2-2a$, and $p^3q+7$. The prefix “bi-” counts terms, not degree; a binomial can have any nonnegative integer degree.
Notation, language, and conditions
Terms are separated by top-level addition or subtraction. Standard form arranges terms by descending degree. A binomial may have a common factor, as $6x^2+9x=3x(2x+3)$, but factoring changes its product representation rather than its expanded two-term classification.
Why this idea matters
A binomial is a two-term polynomial whose term structure determines which factoring identities and multiplication patterns are available.
A dependable method
- Simplify coefficients and combine all like terms.
- Count nonzero top-level terms after simplification.
- Confirm exponents are nonnegative integers and no variable occurs in a denominator.
- If exactly two unlike terms remain, classify the expression as a binomial.
- Name degree and variables separately from term count.
