Binomial

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In algebra, a binomial is a polynomial which is the sum of two terms, which are monomials.[1] It is the simplest kind of polynomial after the monomials.

Definition[edit]

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

a x^n - bx^m \,,

where a and b are numbers, and n and  m are nonnegative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In some contexts, the exponents m and n may be negative, in which case the monomial is a Laurent binomial.

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

a x_1^{n_1}\dotsb x_i^{n_i} - b x_1^{m_1}\dotsb x_i^{m_i}

Some examples of binomials are:

3x - 2x^2
xy + yx^2
x^2 + y^2

Operations on simple binomials[edit]

  • The binomial  x^2 - y^2 can be factored as the product of two other binomials.
 x^2 - y^2 = (x + y)(x - y).
This is a special case of the more general formula:  x^{n+1} - y^{n+1} = (x - y)\sum_{k=0}^{n} x^{k}\,y^{n-k}.
This can also be extended to  x^2 + y^2 = x^2 - (iy)^2 = (x - iy)(x + iy) when working over the complex numbers
  • The product of a pair of linear binomials (ax+b) and (cx+d) is a trinomial:
 (ax+b)(cx+d) = acx^2+(ad+bc)x+bd.
  • 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 x^2+2xy+y^2.
  • 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=2mn, c=n^2+m^2, then a^2+b^2=c^2.

See also[edit]

Notes[edit]

  1. ^ Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011. 
  2. ^ Sturmfels, Bernd (2002). "Solving Systems of Polynomial Equations". CBMS Regional Conference Series in Mathematics (Conference Board of the Mathematical Sciences) (97): 62. Retrieved 21 March 2014. 

References[edit]