A binomial, in the context of algebra, is a polynomial that consists of exactly two distinct terms connected by addition or subtraction. Formally, a binomial can be expressed as
$$ P(x) = a,x^{m} + b,x^{n}, $$
where $a$ and $b$ are non‑zero coefficients belonging to a given coefficient field (commonly the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$), and $m$ and $n$ are non‑negative integers with $m eq n$. The exponents $m$ and $n$ may be equal when the two terms differ only by coefficient, but in such a case the expression simplifies to a monomial; therefore the standard definition of a binomial assumes distinct exponents.
Key properties
- Degree – The degree of a binomial is the larger of $m$ and $n$; i.e., $\deg(P) = \max{m,n}$.
- Factorization – Certain binomials admit factorization formulas, notable among which are:
- Difference of squares: $a^{2} - b^{2} = (a - b)(a + b)$.
- Sum and difference of cubes: $a^{3} \pm b^{3} = (a \pm b)(a^{2} \mp ab + b^{2})$.
- Irreducibility – Over the field of rational numbers $\mathbb{Q}$, a binomial $x^{n} - a$ is irreducible if and only if $a$ is not an $n$th power in $\mathbb{Q}$ and $n$ is prime (Eisenstein’s criterion and rational root tests provide further conditions).
- Binomial theorem – Powers of a binomial give rise to the binomial expansion: $$ (x + y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}, $$ where $\binom{n}{k}$ are binomial coefficients. This theorem underlies many combinatorial and analytic applications.
Examples
- $3x^{4} - 5$ – a binomial with terms of degrees 4 and 0.
- $2x^{2} + 7x$ – a binomial where the degrees differ by one.
- $x^{5} - x^{2}$ – a binomial that can be factored as $x^{2}(x^{3} - 1)$.
Applications
Binomials appear throughout mathematics:
- Algebraic equations – Simple equations such as $ax^{m} + b = 0$ are solved by isolating a single term, often leading to root extraction.
- Number theory – Expressions like $x^{n} \pm 1$ (cyclotomic polynomials) are binomials whose factorization properties relate to the distribution of prime numbers.
- Combinatorics – The coefficients in the expansion of $(x + y)^{n}$ are binomial coefficients, central to counting problems.
- Computer algebra – Algorithms for polynomial factorization frequently treat binomials as base cases due to their simple structure.
Historical notes
The term “binomial” derives from the Latin prefix bi‑ meaning “two” and the Greek nomial (from nomos, “term” or “part”). The concept has been recognized since the development of algebraic notation in the 16th and 17th centuries, and its systematic study was advanced by mathematicians such as Isaac Newton, who formulated the general binomial theorem for arbitrary exponents.