Binomial

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Not to be confused with Binomial distribution.
For other uses, see Binomial (disambiguation).

In algebra, a binomial is a polynomial with two terms[1] —the sum of two monomials—often bound by parentheses or brackets when operated upon. It is the simplest kind of polynomial after the monomials.

Operations on simple binomials

  • The binomial a^{2}-b^{2} can be factored as the product of two other binomials.
a^{2}-b^{2}=(a+b)(a-b).
This is a special case of the more general formula: a^{{n+1}}-b^{{n+1}}=(a-b)\sum _{{k=0}}^{{n}}a^{{k}}\,b^{{n-k}}.
This can also be extended to a^{2}+b^{2}=a^{2}-(ib)^{2}=(a-ib)(a+ib) when working over the complex numbers
  • The product of a pair of linear binomials (ax+b) and (cx+d) is:
(ax+b)(cx+d)=acx^{2}+adx+bcx+bd.
  • A binomial raised to the nth power, represented as
(a+b)^{n}
can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. Taking a simple example, the perfect square binomial (p+q)^{2} can be found by squaring the first term, adding twice the product of the first and second terms and finally adding the square of the second term, to give p^{2}+2pq+q^{2}.
  • A simple but interesting application of the cited binomial formula 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

Notes

  1. ↑ Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011. 

References

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