Crthuky

Prima lex

Leges binominales^[1] in algebra elementaria sunt regulae, per quas possumus multiplicare binomiales. Binomialis (vel binomen) est polynomium duorum membrorum, ut x + 5.

Leges |

Leges binominales solite hae tres sunt:

(a+b)2=a2+2⋅a⋅b+b2displaystyle (a+b)^2=a^2+2cdot acdot b+b^2	Prima Lex Binominalis (Lex Additionis)
(a−b)2=a2−2⋅a⋅b+b2displaystyle (a-b)^2=a^2-2cdot acdot b+b^2	Secunda Lex Binominalis  (Lex Subtractionis)
(a+b)⋅(a−b)=a2−b2displaystyle (a+b)cdot (a-b)=a^2-b^2	Tertia Lex Binominalis (Lex Additionis et Subtractionis)

Probatio Multiplicationis:

(a+b)2=(a+b)⋅(a+b)=a⋅a+a⋅b+b⋅a+b⋅b=a2+2⋅a⋅b+b2displaystyle (a+b)^2=(a+b)cdot (a+b)=acdot a+acdot b+bcdot a+bcdot b=a^2+2cdot acdot b+b^2

(a−b)2=(a−b)⋅(a−b)=a⋅a−a⋅b−b⋅a+b⋅b=a2−2⋅a⋅b+b2displaystyle (a-b)^2=(a-b)cdot (a-b)=acdot a-acdot b-bcdot a+bcdot b=a^2-2cdot acdot b+b^2

(a+b)⋅(a−b)=a⋅a−a⋅b+b⋅a−b⋅b=a2−b2displaystyle (a+b)cdot (a-b)=acdot a-acdot b+bcdot a-bcdot b=a^2-b^2

Et generaliter:

(a+b)⋅(c+d)=a⋅c+a⋅d+b⋅c+b⋅ddisplaystyle (a+b)cdot (c+d)=acdot c+acdot d+bcdot c+bcdot d

Mnemonicon anglicum huius regulae est FOIL, id quod First, Outer, Inner, Last significat:

• First = primi, (a + b)(c + d)


• Outer = externi, (a + b)(c + d)


• Inner = interni, (a + b)(c + d)


• Last = ultimi, (a + b)(c + d)

Latine mementote PEIUs.

Theorema binomiale est generalizatio harum legum.

Bibliographia |

• Bashmakova, I. G., et G. S. Smirnova. The Beginnings and Evolution of Algebra, versio anglica Abe Shenitzer, editor David A. Cox. Vasingtoniae: Mathematical Association of America, 2000. ISBN 0883853299
  


• Kuhn, Harry Waldo. Elementary College Algebra. Novi Eboraci: Macmillan, 1935. OCLC 7634699

Haec stipula ad mathematicam spectat. Amplifica, si potes!