Question

Chứng minh các hằng đẳng thức:

a) a³+b³+c³ - 3.a.b.c = ( a+b+c). ( a²+b²+c² - a.b - b.c - a.c)

b) (a+b+c)³-a³-b³-c³ = 3.(a+b) . (b.c) . (c+a)

Answer 1

a:\(a^3+b^3+c^3-3abc\)

\(=\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc\)

\(=\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)\)

\(=\left(a+b+c\right)\cdot\left(a^2+b^2+c^2-ab-ac-bc\right)\)

b: \(=\left(a+b+c-a\right)\left[\left(a+b+c\right)^2+a\left(a+b+c\right)+a^2\right]-\left(b+c\right)\left(b^2+bc+c^2\right)\)

\(=\left(b+c\right)\left(a^2+2ab+b^2+c^2+2ac+2bc+a^2+ab+ac+a^2-b^2-bc-c^2\right)\)

=3(a+b)(b+c)(a+c)