Question

Chứng minh rằng:

Với a+b+c=0 thì a^4+b^4+c^4=2(ab+bc+ca)^2

Answer 1

+ a + b + c = 0 \(\Rightarrow\left(a+b+c\right)^2=0\)

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

+ \(a^4+b^4+c^4=\left(a^2+b^2+c^2\right)^2-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)

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

\(=4\left(ab+bc+ca\right)^2-2\left[\left(ab+bc+ca\right)^2-2\left(ab^2c+a^2bc+abc^2\right)\right]\)

\(=2\left(ab+bc+ca\right)^2+4\left(ab^2c+abc^2+a^2bc\right)\)

\(=2\left(ab+bc+ca\right)^2+4abc\left(a+b+c\right)\)

\(=2\left(ab+bc+ca\right)^2\)