$$ (a+b)^2 = a^2 +2ab + b^2 $$
$$ (a-b)^2 = a^2 - 2ab + b^2 $$
$$ a^2 - b^2 = (a+b)(a-b) $$
$$ a^2 + b^2 = (a + b)^2 - 2ab $$
$$ a^2 + b^2 = (a - b)^2 + 2ab $$
$$ 2(a+b)^2 = (a + b)^2 + (a - b)^2 $$
$$ ab = (\frac{a+b}{2})^2 - (\frac{a-b}{2})^2$$
$$ 4ab = (a+b)^2 - (a-b)^2 $$
$$ (a+b)^2 = (a-b)^2 + 4ab $$
$$(a-b)^2 = (a+b)^2 - 4ab $$
$$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$
$$(a+b)^3 = a^3+b^3+3ab(a+b) $$
$$(a-b)^3 = a^3 -3a^2b +3ab^2 + b^3 $$
$$ (a-b)^3 = a^3 -b^3 -3ab(a-b)$$
$$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$
$$ a^3 - b^3 = (a-b)(a^2 +ab + b^2) $$
$$ (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab +2bc +2ca $$
$$(a-b-c)^2 = a^2 +b^2+c^2-2ab+2bc-2ca $$
$$a^3 + b^3 + c^3 -3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
$$ If\space (a+b+c) = 0 \space Then \space a^3+b^3+c^3 =0 $$
$$(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 +4ab^3 +b^4 $$
$$(a-b)^4 = a^4 - 4a^3b+6a^2b^2-4ab^3+b^4 $$
$$a^4-b^4 = (a-b)(a+b)(a^2+b^2) $$
$$a^5-b^5 = (a-b)(a^4+a^3b+a^2b^2+ab^3+b^4) $$
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