Question

Chứng minh đẳng thức

a) \(a\left(1-b\right)+a\left(a^2-1\right)=a\left(a^2-6\right)\)

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

Giúp mình với

Answer 1

a ) \(VT=a\left(1-b\right)+a\left(a^2-1\right)\)

\(=a-ab+a^3-a\)

\(=a^3-ab\)

\(=a\left(a^2-b\right)=VP\left(đpcm\right)\)

b ) \(VP=a^3+b^3+c^3-3abc\)

\(=\left(a^3+b^3+3a^2b+3b^2a\right)+c^3-3abc-3a^2b-3b^2a\)

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

\(=\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)\)

\(=\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2-3ab\right]\)

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

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