Question

CMR:\(x^4+y^4+\left(x+y\right)^4=2.\left(x^2+xy+y^2\right)^2\)

Answer 1

Lời giải:

\(x^4+y^4+(x+y)^4-2(x^2+xy+y^2)^2\)

\(=(x^2+y^2)^2-2x^2y^2+[(x+y)^2]^2-2(x^2+xy+y^2)^2\)

\(=(x^2+y^2)^2-(x^2+xy+y^2)^2+(x^2+2xy+y^2)^2-(x^2+xy+y^2)^2-2x^2y^2\)

\(=(x^2+y^2-x^2-xy-y^2)(x^2+y^2+x^2+xy+y^2)+(x^2+2xy+y^2-x^2-xy-y^2)(x^2+2xy+y^2+x^2+xy+y^2)-2x^2y^2\)

\(=-xy(2x^2+xy+2y^2)+xy(2x^2+3xy+2y^2)-2x^2y^2\)

\(=xy(2x^2+3xy+2y^2-2x^2-xy-2y^2-2xy)\)

\(=xy.0=0\)

\(\Rightarrow x^4+y^4+(x+y)^4=2(x^2+xy+y^2)^2\) đpcm