Formule de calcul prescurtat
$a(b ± c) = a \cdot b ± a \cdot c$
$(a + b)^{2} = a^{2} + 2ab + b^{2}$
$(a - b)^{2} = a^{2} - 2ab + b^{2}$
$a^{2} - b^{2} = (a + b)(a - b)$
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac$
$a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})$
$a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})$
$(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$
$(a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}$
Exercitii rezolvate cu formule de calcul prescurtat
$x^{3} + y^{3} + z^{3} - 3xyz = (x + y + z)(x^{2} + y^{2} + z^{2} - xy - yz - zx)$
$x^{3} + y^{3} + z^{3} = (x + y + z)^{3} - 3(x + y)(y + z)(z + x)$
$a^{4} + b^{4} = (a^{2} + b^{2} - ab\sqrt{2})(a^{2} + b^{2} + ab\sqrt{2})$
$a^{4} - b^{4} = (a - b)(a + b)(a^{2} + b^{2})$
$a^{5} + b^{5} = (a + b)(a^{4} - a^{3}b + a^{2}b^{2} - ab^{3} + b^{4})$
$a^{5} - b^{5} = (a - b)(a^{4} + a^{3}b + a^{2}b^{2} + ab^{3} + b^{4})$
$(1 + a)(1 + a^{2} + a^{4}) = 1 + a + a^{2} + a^{3} + a^{4} + a^{5}$
$a^{6} + b^{6} = (a^{3} - 2ab^{2})^{2} + (b^{3} - 2a^{2}b)^{2}$
$a^{n} - b^{n} = (a - b)(a^{n - 1} + a^{n - 2} + ... + ab^{n - 2} + b^{n - 1})$
$a^{2n} - b^{2n} = (a^{2} - b^{2})(a^{2n - 2} + a^{2n - 4}b^{2} + ... + a^{2}b^{2n - 4} + b^{2n - 2})$
$a^{2n + 2} + b^{2n + 2} = (a + b)(a^{2n} + a^{2 - 1}b + ... + ab^{2n - 1} + b^{2n})$
$(1 + a + a^{2} + ... + a^{n})(1 + a^{n + 1}) = 1 + a + a^{2} + a^{3} + ... + a^{2n} + a^{2n + 1}$
Aceste formule se aplica pentru oricare ar fi x, y, z, a, b, c care apartin lui R si n care apartine lui N.