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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

axm−bxn,{displaystyle ax^{m}-bx^{n},,}

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:

ax1n1⋯xini−bx1m1⋯ximi{displaystyle ax_{1}^{n_{1}}dotsb x_{i}^{n_{i}}-bx_{1}^{m_{1}}dotsb x_{i}^{m_{i}}}

Some examples of binomials are:

3x−2x2{displaystyle 3x-2x^{2}}

xy+yx2{displaystyle xy+yx^{2}}

0.9x3+πy2{displaystyle 0.9x^{3}+pi y^{2}}

Operations on simple binomials

• The binomial x² − y² can be factored as the product of two other binomials:

x2−y2=(x+y)(x−y).{displaystyle x^{2}-y^{2}=(x+y)(x-y).}

This is a special case of the more general formula:

xn+1−yn+1=(x−y)∑k=0nxkyn−k.{displaystyle x^{n+1}-y^{n+1}=(x-y)sum _{k=0}^{n}x^{k},y^{n-k}.}

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

x2+y2=x2−(iy)2=(x−iy)(x+iy).{displaystyle x^{2}+y^{2}=x^{2}-(iy)^{2}=(x-iy)(x+iy).}

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

(ax+b)(cx+d)=acx2+(ad+bc)x+bd.{displaystyle (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.}

• A binomial raised to the n^thpower, 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:

(x+y)2=x2+2xy+y2.{displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}

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 = n² − m², b = 2mn, and c = n² + m²; then a² + b² = c².

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

x3+y3=(x+y)(x2−xy+y2){displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}

x3−y3=(x−y)(x2+xy+y2){displaystyle x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}

See also

• Completing the square


• Binomial distribution


• 
  List of factorial and binomial topics (which contains a large number of related links)

Notes

References

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