Binomio

In matematica si definisce binòmio la somma algebrica di due monomi^[1].

\left({a+b}\right)

\left({2x^{3}y-{2 \over 3}z}\right)

Ciascuna lettera, di solito scritta in minuscolo, rappresenta un generico numero reale o complesso.

Potenze di binomio

Il quadrato di un binomio è dato dal quadrato del primo termine, più il doppio prodotto del primo per il secondo, più il quadrato del secondo:^[2]

\left({a+b}\right)^{2}=a^{2}+2ab+b^{2}

Il cubo di un binomio è dato dal cubo del primo monomio, più il triplo del prodotto del quadrato del primo monomio per il secondo, più il triplo del prodotto del primo monomio per il quadrato del secondo, più il cubo del secondo monomio:^[3]

\left({a+b}\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}

\left({a-b}\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}

\left({a+b}\right)^{4}=a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}

Per le potenze di ordine superiore si procede in modo simile:

\left({a+b}\right)^{n}=a^{n}+C_{n,1}a^{n-1}b+C_{n,2}a^{n-2}b^{2}+\dots +C_{n,n-1}ab^{n-1}+b^{n}

dove i $C_{n,1}$ , $C_{n,2}$ , $\dots$ sono i coefficienti binomiali ottenibili tramite il Triangolo di Tartaglia^[4].

Note

^ Bottiroli et al. 2007, p. 326.
^ Bottiroli et al. 2007, p. 334.
^ Bottiroli et al. 2007, p. 335.
^ Bottiroli et al. 2007, p. 337.

Bibliografia

Bruno Bottiroli, Antonio Cantone, Giuliana Pionetti, Corso di Matematica-Algebra 1, Ghisetti e Corvi Editori, 2007, ISBN 978-88-538-0410-5.

Altri progetti

Wikizionario contiene il lemma di dizionario «binomio»
Wikimedia Commons contiene immagini o altri file sul binomio

Collegamenti esterni

Binomio, su Treccani.it – Enciclopedie on line, Istituto dell'Enciclopedia Italiana.
Giovanni Vacca, BINOMIO, in Enciclopedia Italiana, Istituto dell'Enciclopedia Italiana, 1930.
Binòmio, in Dizionario delle scienze fisiche, Istituto dell'Enciclopedia Italiana, 1996.
Binòmio², su Vocabolario Treccani, Istituto dell'Enciclopedia Italiana.
binòmio (sostantivo), su sapere.it, De Agostini.
Binomio, in Enciclopedia della Matematica, Istituto dell'Enciclopedia Italiana, 2013.
(EN) Eric W. Weisstein, Binomial, su MathWorld, Wolfram Research.
(EN) Binomial, su Encyclopaedia of Mathematics, Springer e European Mathematical Society.

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Monomio · Binomio · Trinomio · Polinomio · Prodotti notevoli · Divisione dei polinomi · Divisibilità dei polinomi · Teorema di Ruffini · Regola di Ruffini · Divisibilità di binomi notevoli

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