Binomial 542399 225065426 2008-07-11T18:44:34Z 128.235.72.163 {{otheruses}} In [[elementary algebra]], a '''binomial''' is a [[polynomial]] with two terms: the sum of two [[monomial]]s. It is the simplest kind of polynomial except for a monomial. The binomial <math> a^2 - b^2 </math> can be factored as the product of two other binomials: :<math> a^2 - b^2 = (a + b)(a - b). </math> (This is a special case of the more general formula <math> a^{n+1} - b^{n+1} = (a - b)\sum_{k=0}^{n} a^{k}\,b^{n-k}</math>.) The product of a pair of linear binomials ''a x + b'' and ''c x + d'' is: :<math> (a x + b)(c x + d) = a c x^2 + (a d + b c) x + b d.</math> A binomial raised to the ''n<sup>th</sup>'' [[Exponentiation|power]], represented as :<math> (a + b)^n </math> can be expanded by means of the [[binomial theorem]] or, equivalently, using [[Pascal's triangle]]. ===Example=== A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating [[Pythagorean triple]]s: for ''m < n'', let <math>a=n^2-m^2,\ b=2mn,\ c=n^2+m^2</math>, then <math>a^2+b^2=c^2</math>. == See also == *[[Completing the square]] *[[Binomial distribution]] *[[Binomial coefficient]] * [[Binomial-QMF]] (Daubechies Wavelet Filters) *The [[list of factorial and binomial topics]] contains a large number of related links. == External links == * [http://www.allmathwords.org/binomial.html Binomial] All Math Words Encyclopedia for grades 7-10. [[Category:Algebra]] [[ca:Binomi]] [[he:נוסחאות הכפל המקוצר]] [[de:Binom]] [[et:Binoom]] [[es:Igualdades notables]] [[eo:Vikipedio:Projekto matematiko/Dutermo]] [[fr:Binôme (mathématique)]] [[it:Binomio]] [[pl:Dwumian]] [[sl:Binom]] [[sv:Binom]] [[fi:Binomi]] [[uk:Біном]] [[zh:二项式]] [[nl:Merkwaardig product]]