The title of this article is ambiguous. For other meanings see
binomials .
A binomial ( Latin bi “two”; noun “name”) is a polynomial with two terms in mathematics . More precisely: A binomial is the sum of two monomials . For example are
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{\ displaystyle a + b, \ x- \ pi, \ x ^ {2} + y ^ {2}, \ 3ab ^ {5} -4c ^ {3}, \ {\ tfrac {p ^ {2}} {2}} - q}
Binomials. The term is not a binomial, but the square of a binomial.
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{\ displaystyle (a + b) ^ {2}}
The term "binomial" goes back to Euclid.
Calculation rules
The following rules apply to the multiplication of two binomials using the associative and distributive law :
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{\ displaystyle (a + b) (c + d) = ac + ad + bc + bd}
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{\ displaystyle (a + b) (cd) = ac-ad + bc-bd}
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{\ displaystyle (ab) (cd) = ac-ad-bc + bd}
In verbal terms: Multiply every term of the first binomial (the first bracket) by every term of the second binomial (the second bracket).
The following special cases are known as binomial formulas :
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{\ displaystyle (a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2}}
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{\ displaystyle (ab) ^ {2} = a ^ {2} -2ab + b ^ {2}}
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{\ displaystyle (a + b) (ab) = a ^ {2} -b ^ {2}}
The binomial theorem provides a representation for arbitrarily high powers of a binomial:
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{\ displaystyle (a + b) ^ {n} = \ sum _ {k = 0} ^ {n} {n \ choose k} a ^ {nk} b ^ {k}}
The coefficients are called binomial coefficients and can be defined by this formula.
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{\ displaystyle {\ tbinom {n} {k}}}
See also
Web links
Individual evidence
^ Barth, Federle, Haller: Algebra 1 . Ehrenwirth-Verlag, Munich 1980, p. 187, footnote **, there explanation of the designation binomial formula: "In book X of his elements Euclid names a two-part sum ἐκ δύο ὀνομάτων (ek dýo onomáton), consisting of two names."
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{\ displaystyle a + b}
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">