Binomo formulė – dažnai dar vadinama Niutono formule , yra svarbi matematikos teorema, padedanti rasti dvinario , pakelto n -tuoju laipsniu, skleidinį. Teorema dažniausiai yra užrašoma
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{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}}
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{\displaystyle (a+b)^{n}={n \choose 0}a^{n}+{n \choose 1}a^{n-1}b+\dots +{n \choose k}a^{n-k}b^{k}+\dots +{n \choose n}b^{n}}
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{\displaystyle {n \choose k}={n! \over {k!\cdot (n-k)!}}=C_{n}^{k}}
Paskalio trikampio eilutės.
Arba
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{\displaystyle (a+b)^{n}=C_{n}^{0}a^{n}b^{0}+C_{n}^{1}a^{n-1}b^{1}+C_{n}^{2}a^{n-2}b^{2}+C_{n}^{3}a^{n-3}b^{3}+...+C_{n}^{m}a^{n-m}b^{m}+...+C_{n}^{n-1}a^{1}b^{n-1}+C_{n}^{n}a^{0}b^{n},}
kur
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{\displaystyle C_{n}^{k}}
deriniai . Jei
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{\displaystyle (a-b)^{n}}
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{\displaystyle (a-b)^{5}=C_{5}^{0}a^{5}-C_{5}^{1}a^{4}b+C_{5}^{2}a^{3}b^{2}-C_{5}^{3}a^{2}b^{3}+C_{5}^{4}ab^{4}-C_{5}^{5}b^{5}}
Niutono formulė gali būti užrašyta dar taip:
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{\displaystyle (a+b)^{n}=a^{n}+na^{n-1}b+{\frac {n(n-1)}{2!}}a^{n-2}b^{2}+{\frac {n(n-1)(n-2)}{3!}}a^{n-3}b^{3}+...+{\frac {n(n-1)(n-2)...(n-m+1)}{m!}}a^{n-m}b^{m}+...+nab^{n-1}+b^{n}.}
Skirtumo laipsnis:
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{\displaystyle (a-b)^{n}=a^{n}-na^{n-1}b+{\frac {n(n-1)}{2!}}a^{n-2}b^{2}-{\frac {n(n-1)(n-2)}{3!}}a^{n-3}b^{3}+...+(-1)^{m}{\frac {n(n-1)(n-2)...(n-m+1)}{m!}}a^{n-m}b^{m}+...+(-1)^{n}b^{n}.}
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{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
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{\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}
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{\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}}
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{\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}}
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{\displaystyle 0!=1.}
Penktos eilės Niutono binomo formulė yra tokia:
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{\displaystyle (a-b)^{5}=C_{5}^{0}a^{5}-C_{5}^{1}a^{4}b+C_{5}^{2}a^{3}b^{2}-C_{5}^{3}a^{2}b^{3}+C_{5}^{4}ab^{4}-C_{5}^{5}b^{5}=}
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{\displaystyle ={5! \over {0!\cdot (5-0)!}}a^{5}-{5! \over {1!\cdot (5-1)!}}a^{4}b+{5! \over {2!\cdot (5-2)!}}a^{3}b^{2}-{5! \over {3!\cdot (5-3)!}}a^{2}b^{3}+{5! \over {4!\cdot (5-4)!}}ab^{4}-{5! \over {5!\cdot (5-5)!}}b^{5}=}
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{\displaystyle ={5! \over {1\cdot 5!}}a^{5}-5a^{4}b+{5! \over {2!\cdot 3!}}a^{3}b^{2}-{5\cdot 4\cdot 3 \over {3\cdot 2}}a^{2}b^{3}+{5! \over 4!}ab^{4}-{5! \over {5!\cdot 0!}}b^{5}=}
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{\displaystyle =a^{5}-5a^{4}b+{5\cdot 4 \over 2}a^{3}b^{2}-5\cdot 2a^{2}b^{3}+5ab^{4}-b^{5}=}
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{\displaystyle =a^{5}-5a^{4}b+10a^{3}b^{2}-10a^{2}b^{3}+5ab^{4}-b^{5}.}
Užrašysime ketvirto laipsnio Niutono binomo formulę:
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{\displaystyle (a-b)^{4}=C_{4}^{0}a^{4}b^{0}-C_{4}^{1}a^{3}b+C_{4}^{2}a^{2}b^{2}-C_{4}^{3}ab^{3}+C_{4}^{4}a^{0}b^{4}=}
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{\displaystyle ={\frac {4!}{0!(4-0)!}}a^{4}b^{0}-{\frac {4!}{1!(4-1)!}}a^{3}b+{\frac {4!}{2!(4-2)!}}a^{2}b^{2}-{\frac {4!}{3!(4-3)!}}ab^{3}+{\frac {4!}{4!(4-4)!}}a^{0}b^{4}=}
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{\displaystyle ={\frac {4!}{4!}}a^{4}-{\frac {4!}{3!}}a^{3}b+{\frac {4!}{2!\cdot 2!}}a^{2}b^{2}-{\frac {4!}{3!}}ab^{3}+{\frac {4!}{4!}}b^{4}=}
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{\displaystyle =a^{4}-4a^{3}b+{\frac {4\cdot 3\cdot 2}{4}}a^{2}b^{2}-4ab^{3}+b^{4}=}
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{\displaystyle =a^{4}-4a^{3}b+6a^{2}b^{2}-4ab^{3}+b^{4}.}
Patikriname, kai a=6, b=2,
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{\displaystyle (a-b)^{4}=(6-2)^{4}=4^{4}=256;}
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{\displaystyle a^{4}-4a^{3}b+6a^{2}b^{2}-4ab^{3}+b^{4}=6^{4}-4\cdot 6^{3}\cdot 2+6\cdot 6^{2}\cdot 2^{2}-4\cdot 6\cdot 2^{3}+2^{4}=}
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{\displaystyle =1296-4\cdot 216\cdot 2+6\cdot 36\cdot 4-4\cdot 6\cdot 8+16=1296-1728+864-192+16=256.}
