[latex](a+b+c)^3=(a+b+c)(a+b+c)(a+b+c)=[/latex] [latex](a^2+ab+ac+ab+b^2+bc+ac+bc+c^2)(a+b+c)=[/latex] [latex](a^2 + 2ab + 2ac + b^2 + 2bc + c^2)(a+b+c)=[/latex] [latex]a^3+a^2b+a^2c+2a^2b+2ab^2+2abc+2a^2c+2abc+2ac^2+ab^2+b^3+b^2c+2abc+2b^2c+2bc^2+ac^2+bc^2+c^3=[/latex] [latex]a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3[/latex] ========================== Wykaż że: [latex](a+b+c)^3+(a-b-c)^3+(c-a-b)^3+(b-a-c)^3=24abc[/latex] ----------------- [latex](a+b+c)^3=a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3[/latex] ----------------- [latex](a-b-c)^3=a^3 + 3a^2(-b) + 3a^2(-c) + 3a(-b)^2 + 6a(-b)(-c) + 3a(-c)^2 + (-b)^3 + 3(-b)^2(-c) + 3(-b)(-c)^2 + (-c)^3=[/latex] [latex]a^3 - 3a^2b - 3a^2c + 3ab^2 + 6abc + 3ac^2 - b^3 - 3b^2c - 3bc^2 - c^3[/latex] ----------------- [latex](c-a-b)^3=(-a-b+c)^3=(-a)^3 + 3(-a)^2(-b) + 3(-a)^2c + 3(-a)(-b)^2 + 6(-a)(-b)c + 3(-a)c^2 + (-b)^3 + 3(-b)^2c + 3(-b)c^2 + c^3=[/latex] [latex]- a^3 - 3a^2b + 3a^2c - 3ab^2 + 6abc - 3ac^2 - b^3 + 3b^2c - 3bc^2 + c^3[/latex] ----------------- [latex](b-a-c)^3=(-a+b-c)^3=(-a)^3 + 3(-a)^2b + 3(-a)^2(-c) + 3(-a)b^2 + 6(-a)b(-c) + 3(-a)(-c)^2 + b^3 + 3b^2(-c) + 3b(-c)^2 + (-c)^3=[/latex] [latex]- a^3 + 3a^2b - 3a^2c - 3ab^2 + 6abc - 3ac^2 + b^3 - 3b^2c + 3bc^2 - c^3[/latex] ----------------- dodajemy stronami: [latex](a+b+c)^3+(a-b-c)^3+(c-a-b)^3+(b-a-c)^3=24abc[/latex]
