Question

Cho a +b + c +d =0 

C/m : \(a^3+b^3+c^3+d^3=3(c+d)(ab-cd)\)

Answer 1

Ta có : \(a+b+c+d=0\)

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

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

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

\(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)+c^3+d^3+3cd\left(c+d\right)=0\)( chỗ này mình làm hơi tắt bỏ qua nha )

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

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

Mà theo ta có : \(a+b=-\left(c+d\right)\)

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