Question

Cho a+b+c=0

Chứng minh rằng:\(2\left(a^5+b^5+c^5\right)=5abc\left(a^2+b^2+c^2\right)\)

Mysterious PersonHung nguyen

Answer 1

Ta có:

\(a+b+c=0\)

\(\Leftrightarrow\left(a+b\right)^5=\left(-c\right)^5\)

\(\Leftrightarrow a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5=-c^5\)

\(\Leftrightarrow a^5+b^5+c^5=-5ab\left(a^3+2a^2b+2ab^2+b^3\right)\)

\(\Leftrightarrow a^5+b^5+c^5=-5ab\left[\left(a+b\right)\left(a^2-ab+b^2\right)+2ab\left(a+b\right)\right]\)

\(\Leftrightarrow a^5+b^5+c^5=5abc\left(a^2+ab+b^2\right)\)

\(\Leftrightarrow2\left(a^5+b^5+c^5\right)=5abc\left[a^2+b^2+\left(a^2+2ab+b^2\right)\right]\)

\(\Leftrightarrow2\left(a^5+b^5+c^5\right)=5abc\left(a^2+b^2+c^2\right)\)