“Binomial (polynomial)”的意思、由来-开放百科全书

词条

Binomial (polynomial)

释义

Definition
Operations on simple binomials
See also
Notes
References

{{other uses|Binomial (disambiguation){{!}}Binomial}}

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.

Definition

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 {{math|a}} and {{math|b}} are numbers, and {{math|m}} and {{math|n}} are distinct nonnegative integers and {{math|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 {{math|m}} and {{math|n}} may be negative.

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

Some examples of binomials are:

Operations on simple binomials

The binomial {{math|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 {{math|(ax + b)}} and {{math|(cx + d)}} is a trinomial:

A binomial raised to the {{math|n}}^th power, represented as {{math|(x + y)ⁿ}} can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square {{math|(x + y)²}} of the binomial {{math|(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 {{math|n}}^th power uses the numbers {{math|n}} rows down from the top of the triangle.

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

For {{math|m < n}}, let {{math|a {{=}} n² − m²}}, {{math|b {{=}} 2mn}}, and {{math|c {{=}} n² + m²}}; then {{math|a² + b² {{=}} c²}}.

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

Notes

1. ^{{Cite web | last = Weisstein | first = Eric | authorlink = Eric Weisstein | coauthors = | title = Binomial | work = | publisher = Wolfram MathWorld | date = | url = http://mathworld.wolfram.com/Binomial.html | doi = | accessdate = 29 March 2011}}
2. ^{{Cite journal | last = Sturmfels | first = Bernd | authorlink = Bernd Sturmfels | journal = CBMS Regional Conference Series in Mathematics | title = Solving Systems of Polynomial Equations | publisher = Conference Board of the Mathematical Sciences | issue = 97 | page = 62 | year = 2002 | url = https://books.google.com/books?id=N9c8bWxkz9gC | accessdate = 21 March 2014}}

References

{{cite book |first1=L. |last1=Bostock |author-link1=Linda Bostock |first2=S. |last2=Chandler |author-link2=Sue Chandler |title=Pure Mathematics 1 |isbn=0-85950-092-6 |publisher=Oxford University Press |date=1978 |page=36}}

2 : Algebra|Factorial and binomial topics

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Definition

Operations on simple binomials

See also

Notes

References