In algebra, a **binomial** is a polynomial that is the sum of two terms, each of which is a monomial.^{ [1] } It is the simplest kind of polynomial after the monomials.

A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form

where *a* and *b* are numbers, and *m* and *n* are distinct nonnegative integers and *x* is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a *Laurent binomial*, often simply called a *binomial*, is similarly defined, but the exponents *m* and *n* may be negative.

More generally, a binomial may be written^{ [2] } as:

- The binomial
*x*^{2}−*y*^{2}can be factored as the product of two other binomials:

- This is a special case of the more general formula:
- When working over the complex numbers, this can also be extended to:

- The product of a pair of linear binomials (
*ax*+*b*) and (*cx*+*d*) is a trinomial:

- A binomial raised to the
*n*^{th}power, represented as (*x + y*)^{n}can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square (*x + y*)^{2}of the binomial (*x + y*) is equal to the sum of the squares of the two terms and twice the product of the terms, that is:

- The numbers (1, 2, 1) appearing as multipliers for the terms in this expansion are binomial coefficients two rows down from the top of Pascal's triangle. The expansion of the
*n*^{th}power uses the numbers*n*rows down from the top of the triangle.

- An application of above formula for the square of a binomial is the "(
*m, n*)-formula" for generating Pythagorean triples:

- For
*m < n*, let*a*=*n*^{2}−*m*^{2},*b*= 2*mn*, and*c*=*n*^{2}+*m*^{2}; then*a*^{2}+*b*^{2}=*c*^{2}.

- Binomials that are sums or differences of cubes can be factored into lower-order polynomials as follows:

- Completing the square
- Binomial distribution
- List of factorial and binomial topics (which contains a large number of related links)

- ↑ Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011.CS1 maint: discouraged parameter (link)
- ↑ Sturmfels, Bernd (2002).
*Solving Systems of Polynomial Equations*.*CBMS Regional Conference Series in Mathematics*.**97**. American Mathematical Society. p. 62. ISBN 9780821889411.

In mathematics, the **binomial coefficients** are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers *n* ≥ *k* ≥ 0 and is written It is the coefficient of the *x*^{k} term in the polynomial expansion of the binomial power (1 + *x*)^{n}, and is given by the formula

In elementary algebra, the **binomial theorem** describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (*x* + *y*)^{n} into a sum involving terms of the form *ax*^{b}*y*^{c}, where the exponents b and c are nonnegative integers with *b* + *c* = *n*, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

In mathematics, **Pascal's triangle** is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

In mathematics, a **coefficient** is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. When variables appear in the coefficients, they are often called parameters, and must be clearly distinguished from those representing other variables in an expression.

In mathematics, the **discriminant** of a polynomial is a quantity that depends on the coefficients and determines various properties of the roots. It is generally defined as a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. It is often denoted by the symbol .

In mathematics, **factorization** or **factoring** consists of writing a number or another mathematical object as a product of several *factors*, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (*x* – 2)(*x* + 2) is a factorization of the polynomial *x*^{2} – 4.

**Exponentiation** is a mathematical operation, written as *b*^{n}, involving two numbers, the *base*b and the *exponent* or *power*n, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, *b*^{n} is the product of multiplying n bases:

In algebra, a **quadratic function**, a **quadratic polynomial**, a **polynomial of degree 2**, or simply a **quadratic**, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.

In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of **binomial type** if it satisfies the sequence of identities

In mathematics, a **Sheffer sequence** or **poweroid** is a polynomial sequence, i.e., a sequence {*p*_{n}(*x*) : *n* = 0, 1, 2, 3, ... } of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.

In elementary algebra, **completing the square** is a technique for converting a quadratic polynomial of the form

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a **Gröbner basis** is a particular kind of generating set of an ideal in a polynomial ring *K*[*x*_{1}, …, *x*_{n}] over a field *K*. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.

In mathematics, an **algebraic equation** or **polynomial equation** is an equation of the form

In elementary algebra, * FOIL* is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the

In mathematics, **Descartes' rule of signs**, first described by René Descartes in his work *La Géométrie*, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients, and that the difference between these two numbers is always even. This implies, in particular, that if the number of sign changes is zero or one, then there are exactly zero or one positive roots, respectively.

In mathematics, an **expansion** of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a sum of (repeated) products. During the expansion, simplifications such as grouping of like terms or cancellations of terms may also be applied. Instead of multiplications, the expansion steps could also involve replacing powers of a sum of terms by the equivalent expression obtained from the binomial formula; this is a shortened form of what would happen if the power were treated as a repeated multiplication, and expanded repeatedly. It is customary to reintroduce powers in the final result when terms involve products of identical symbols.

**Zhegalkin****polynomials** are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, *x*^{2} = *x*. Hence a polynomial such as 3*x*^{2}*y*^{5}*z* is congruent to, and can therefore be rewritten as, *xyz*.

In mathematics, in the subfield of ring theory, a ring *R* is a **polynomial identity ring** if there is, for some *N* > 0, an element *P* other than 0 of the free algebra, Z⟨*X*_{1}, *X*_{2}, ..., *X*_{N}⟩, over the ring of integers in *N* variables *X*_{1}, *X*_{2}, ..., *X*_{N} such that for all *N*-tuples *r*_{1}, *r*_{2}, ..., *r*_{N} taken from *R* it happens that

In abstract algebra, a **monomial ideal** is an ideal generated by monomials in a multivariate polynomial ring over a field.

- Bostock, L.; Chandler, S. (1978).
*Pure Mathematics 1*. Oxford University Press. p. 36. ISBN 0-85950-092-6.

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