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Concept
Binomial (polynomial)
Summary
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse 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 non-negative 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 as:
The binomial x2 − y2 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 nth 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 the binomial coefficients two rows down from the top of Pascal's triangle. The expansion of the nth power uses the numbers n rows down from the top of the triangle.
An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating Pythagorean triples:
For m < n, let a = n2 − m2, b = 2mn, and c = n2 + m2; then a2 + b2 = c2.
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Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. with variables with variables with variables the quadratic polynomial in standard form with variables. with variables, nonnegative integers and any constants. where is variable and constants are nonnegative integers and any constants. A trinomial equation is a polynomial equation involving three terms. An example is the equation studied by Johann Heinrich Lambert in the 18th century.
Algebra
Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) 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 axbyc, 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, for n = 4, The coefficient a in the term of axbyc is known as the binomial coefficient or (the two have the same value).