Leges binominales

Prima lex

Leges binominales 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:

$(a+b{)}^2=a^2+2\cdot a\cdot b+b^2$	Prima Lex Binominalis (Lex Additionis)
$(a-b{)}^2=a^2-2\cdot a\cdot b+b^2$	Secunda Lex Binominalis  (Lex Subtractionis)
$(a+b)\cdot (a-b)=a^2-b^2$	Tertia Lex Binominalis (Lex Additionis et Subtractionis)

Probatio Multiplicationis:

$(a+b{)}^2=(a+b)\cdot (a+b)=a\cdot a+a\cdot b+b\cdot a+b\cdot b=a^2+2\cdot a\cdot b+b^2$

$(a-b{)}^2=(a-b)\cdot (a-b)=a\cdot a-a\cdot b-b\cdot a+b\cdot b=a^2-2\cdot a\cdot b+b^2$

$(a+b)\cdot (a-b)=a\cdot a-a\cdot b+b\cdot a-b\cdot b=a^2-b^2$

Et generaliter:

$(a+b)\cdot (c+d)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$

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