Question

1) Cho a + b - c = 0 . Chứng minh rằng : a³ + b³ - c³ = -3abc

2) Cho a - b + c = 0 . Chứng minh rằng : a³ - b³ + c³ = -3abc

Các bạn giải gấp cho mình 2 câu này nha . Mình đag cần gấp .

Answer 1

1 ) Ta có :

\(a+b-c=0\Leftrightarrow a+b=c\Leftrightarrow\left(a+b\right)^3=c^3\)

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

\(\Rightarrow a^3+b^3-c^3=a^3+b^3-3a^2b-3b^2a-b^3\)

\(\Rightarrow a^3+b^3-c^3=-3a^2b-3b^2a\)

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

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

2 ) Ta có :

\(a-b+c=0\Leftrightarrow c=b-a\Leftrightarrow c^3=\left(b-a\right)^3\)

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

\(\Rightarrow a^3-b^3+c^3=a^3-b^3+b^3-3a^2b+3b^2a-a^3\)

\(\Rightarrow a^3-b^3+c^3=-3a^2b+3b^2a\)

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

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

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

Answer 2

1 ) Bổ sung dấu \(\Rightarrow\) thứ 2 :

\(\Rightarrow...=a^3+b^3-a^3-3a^2b-3b^2a-b^3\)

Answer 3

Làm lại 2) :

\(a-b+c=0\Leftrightarrow c=b-a\Leftrightarrow c^3=\left(b-a\right)^3\)

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

\(\Rightarrow a^3-b^3+c^3=a^3-b^3+b^3-3b^2a+3ab^2-a^3\)

\(\Rightarrow a^3-b^3+c^3=-3b^2a+3ab^2\)

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