Binomo formulė

Binomo formulė – dažnai dar vadinama Niutono formule, – svarbi matematikos teorema, padedanti rasti dvinario, pakelto n-tuoju laipsniu, skleidinį. Teorema dažniausiai yra užrašoma

${\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}=\sum _{k=0}^{n}C_{n}^{k}a^{n-k}b^{k}}$

arba

${\displaystyle (a+b)^{n}={n \choose 0}a^{n}+{n \choose 1}a^{n-1}b+\dots +{n \choose k}a^{n-k}b^{k}+\dots +{n \choose n}b^{n}}$

Dešinioji šios formulės dalis vadinama binomo laipsnio dėstiniu.^[1]

Skaičiai ${\displaystyle {n \choose k}=C_{n}^{k}={n! \over {k!\cdot (n-k)!}}}$ yra vadinami binomo koeficientais ir yra lygūs skaičiams iš atitinkamos Paskalio trikampio eilutės.

arba

${\displaystyle (a+b)^{n}=C_{n}^{0}a^{n}b^{0}+C_{n}^{1}a^{n-1}b^{1}+C_{n}^{2}a^{n-2}b^{2}+\dots +C_{n}^{n-1}a^{1}b^{n-1}+C_{n}^{n}a^{0}b^{n}}$

kur ${\displaystyle C_{n}^{k}}$ yra deriniai. Jei ${\displaystyle (a-b)^{n}}$, tada bus tai minusas tai pliusas, pradedant nuo minuso, pvz.:

${\displaystyle (a-b)^{5}=C_{5}^{0}a^{5}-C_{5}^{1}a^{4}b+C_{5}^{2}a^{3}b^{2}-C_{5}^{3}a^{2}b^{3}+C_{5}^{4}ab^{4}-C_{5}^{5}b^{5}}$

• ${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$
• ${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}$
• ${\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}}$
• ${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}}$

Pastaba, ${\displaystyle 0!=1.}$

Penktos eilės Niutono binomo formulė yra tokia:

${\displaystyle (a-b)^{5}=C_{5}^{0}a^{5}-C_{5}^{1}a^{4}b+C_{5}^{2}a^{3}b^{2}-C_{5}^{3}a^{2}b^{3}+C_{5}^{4}ab^{4}-C_{5}^{5}b^{5}=}$

${\displaystyle ={5! \over {0!\cdot (5-0)!}}a^{5}-{5! \over {1!\cdot (5-1)!}}a^{4}b+{5! \over {2!\cdot (5-2)!}}a^{3}b^{2}-{5! \over {3!\cdot (5-3)!}}a^{2}b^{3}+{5! \over {4!\cdot (5-4)!}}ab^{4}-{5! \over {5!\cdot (5-5)!}}b^{5}=}$

${\displaystyle ={5! \over {1\cdot 5!}}a^{5}-5a^{4}b+{5! \over {2!\cdot 3!}}a^{3}b^{2}-{5\cdot 4\cdot 3 \over {3\cdot 2}}a^{2}b^{3}+{5! \over 4!}ab^{4}-{5! \over {5!\cdot 0!}}b^{5}=}$

${\displaystyle =a^{5}-5a^{4}b+{5\cdot 4 \over 2}a^{3}b^{2}-5\cdot 2a^{2}b^{3}+5ab^{4}-b^{5}=}$

${\displaystyle =a^{5}-5a^{4}b+10a^{3}b^{2}-10a^{2}b^{3}+5ab^{4}-b^{5}.}$

Užrašysime ketvirto laipsnio Niutono binomo formulę:

${\displaystyle (a-b)^{4}=C_{4}^{0}a^{4}b^{0}-C_{4}^{1}a^{3}b+C_{4}^{2}a^{2}b^{2}-C_{4}^{3}ab^{3}+C_{4}^{4}a^{0}b^{4}=}$

${\displaystyle ={\frac {4!}{0!(4-0)!}}a^{4}b^{0}-{\frac {4!}{1!(4-1)!}}a^{3}b+{\frac {4!}{2!(4-2)!}}a^{2}b^{2}-{\frac {4!}{3!(4-3)!}}ab^{3}+{\frac {4!}{4!(4-4)!}}a^{0}b^{4}=}$

${\displaystyle ={\frac {4!}{4!}}a^{4}-{\frac {4!}{3!}}a^{3}b+{\frac {4!}{2!\cdot 2!}}a^{2}b^{2}-{\frac {4!}{3!}}ab^{3}+{\frac {4!}{4!}}b^{4}=}$

${\displaystyle =a^{4}-4a^{3}b+{\frac {4\cdot 3\cdot 2}{4}}a^{2}b^{2}-4ab^{3}+b^{4}=}$

${\displaystyle =a^{4}-4a^{3}b+6a^{2}b^{2}-4ab^{3}+b^{4}.}$

Taip pat skaitykite

• Paskalio trikampis
• Binominis skirstinys
• Deriniai