Question

Chứng minh rằng nếu a+b+c=0

Thì 2(a^{5​​ ​}+ b^{5​ ​​​}+ c⁵) = 5abc ( a² + b² + c²)

Answer 1

Giải:

Ta có:

\(a+b+c=0\Leftrightarrow a+b=-c\Leftrightarrow\left(a+b\right)^5=\left(-c\right)^5\)

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

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

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

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

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