Английская Википедия:Binomial (polynomial)

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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.

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

<math>a x^m - bx^n ,</math>

where Шаблон:Math and Шаблон:Math are numbers, and Шаблон:Math and Шаблон:Math are distinct non-negative integers and Шаблон:Math 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 Шаблон:Math and Шаблон:Math may be negative.

More generally, a binomial may be written as:

<math>a\, x_1^{n_1}\dotsb x_i^{n_i} - b\, x_1^{m_1}\dotsb x_i^{m_i}</math>

Examples

<math>3x - 2x^2</math>

<math>xy + yx^2</math>

<math>0.9 x^3 + \pi y^2</math>

<math>2 x^3 + 7</math>

Operations on simple binomials

• The binomial Шаблон:Math can be factored as the product of two other binomials:

<math> x^2 - y^2 = (x - y)(x + y). </math>

This is a special case of the more general formula:

<math> x^{n+1} - y^{n+1} = (x - y)\sum_{k=0}^{n} x^{k} y^{n-k}.</math>

When working over the complex numbers, this can also be extended to:

<math> x^2 + y^2 = x^2 - (iy)^2 = (x - iy)(x + iy). </math>

• The product of a pair of linear binomials Шаблон:Math and Шаблон:Math is a trinomial:

<math> (ax+b)(cx+d) = acx^2+(ad+bc)x+bd.</math>

• A binomial raised to the Шаблон:Math^th power, represented as Шаблон:Math can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square Шаблон:Math of the binomial Шаблон:Math is equal to the sum of the squares of the two terms and twice the product of the terms, that is:

<math> (x + y)^2 = x^2 + 2xy + y^2.</math>

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 Шаблон:Math^th power uses the numbers Шаблон:Math rows down from the top of the triangle.

• An application of the above formula for the square of a binomial is the "Шаблон:Math-formula" for generating Pythagorean triples:

For Шаблон:Math, let Шаблон:Math, Шаблон:Math, and Шаблон:Math; then Шаблон:Math.

• Binomials that are sums or differences of cubes can be factored into smaller-degree polynomials as follows:

<math> x^3 + y^3 = (x + y)(x^2 - xy + y^2) </math>

<math> x^3 - y^3 = (x - y)(x^2 + xy + y^2) </math>

See also

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

References

• Шаблон:Cite book