Draft:Remarkable identity

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Last edited by Bearcat (talk | contribs) 5 months ago. (Update)

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Squares of expression

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

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

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

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

Calculating Cube expressions

$a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})$

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

$(a\pm b)^{3}=a^{3}\pm 3a^{2}b+3ab^{2}\pm b^{3}$

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

Four-degree expression

$(a\pm b)^{4}=a^{4}\pm 4a^{3}b+6a^{2}b^{2}\pm 4ab^{3}+b^{4}$

$(a+b)^{4}=a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}$
$(a-b)^{4}=a^{4}-4a^{3}b+6a^{2}b^{2}-4ab^{3}+b^{4}$

$a^{4}-b^{4}=(a^{2}-b^{2})(a^{2}+b^{2})$

N degree expression

$a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+...+a^{2}b^{n-3}+ab^{n-2}+b[\log _{10}n-1-]$
$a^{2n}-b^{2n}=(a+b)(a^{2n-1}-a^{2n-2}b+a^{2n-3}b^{2}-...-a^{2}b^{2n-3}+ab^{2n-2}-b^{2n-1})$ , burada $n\in N$
$a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}+a^{2n-1}b+a^{2n-2}b^{2}-...-a^{2}b^{2n-2}+ab^{2n-1}-b^{2n})$ , burada $n\in N$

with this type of attribute