Назад
ГДЗ Алгебра 7 клас Мерзляк (2024)
§ 1. Алгебраїчні вирази
Реклама:
Доведіть тотожність:
1) (a + b + c)3 – a3 – b3 – c3 = ((a + b + c)3 – a3) – (b3 + c3) = (a + b + c – a)((a + b + c)2 + (a + b + c)a + a2) – (b + c)(b2 – bc + c2) = (b + c)(a2 + b2 + c2 + 2ab + 2ac + 2bc + a2 + ab + ac + a2) – (b + c)(3a2 + 3ab + 3ac + 3bc) = 3(b + c)(a2 + ab + ac + bc) = 3(b + c)(a(a + b) + c(a + b)) = 3(a + b)(b + c)(a + c);
2) (a – b)3 + (b – c)3 – (a – c)3 = (a – b + b – c)((a – b)2 – (a – b)(b – c) + (b – c)2) – (a – c)3 = (a – c)((a – b)2 – (a – b)(b – c) + (b – c)2) – (a – c)3 = (a – c)((a – b)2 – (a – b)(b – c) + (b – c)2 – (a – c)2) = (a – c)(a2 – 2ab + b2 – ab + ac + b2 – bc + b2 – 2bc + c2 – a2 + 2ac – c2) = (a – c) ∙ (–3ab + 3b2 + 3ac – 3bc) = (a – c)(–3b(a – b) + 3c(a – b)) = (a – c)(a – b)(–3b + 3c) = –3(a – b)(b – c)(a – c).
1) (a + b + c)3 – a3 – b3 – c3 = ((a + b + c)3 – a3) – (b3 + c3) = (a + b + c – a)((a + b + c)2 + (a + b + c)a + a2) – (b + c)(b2 – bc + c2) = (b + c)(a2 + b2 + c2 + 2ab + 2ac + 2bc + a2 + ab + ac + a2) – (b + c)(3a2 + 3ab + 3ac + 3bc) = 3(b + c)(a2 + ab + ac + bc) = 3(b + c)(a(a + b) + c(a + b)) = 3(a + b)(b + c)(a + c);
2) (a – b)3 + (b – c)3 – (a – c)3 = (a – b + b – c)((a – b)2 – (a – b)(b – c) + (b – c)2) – (a – c)3 = (a – c)((a – b)2 – (a – b)(b – c) + (b – c)2) – (a – c)3 = (a – c)((a – b)2 – (a – b)(b – c) + (b – c)2 – (a – c)2) = (a – c)(a2 – 2ab + b2 – ab + ac + b2 – bc + b2 – 2bc + c2 – a2 + 2ac – c2) = (a – c) ∙ (–3ab + 3b2 + 3ac – 3bc) = (a – c)(–3b(a – b) + 3c(a – b)) = (a – c)(a – b)(–3b + 3c) = –3(a – b)(b – c)(a – c).