Ta có:
\(VT=2\left(x^2+xy+y^2\right)^2\)
\(=2\left[\left(x^2\right)^2+\left(xy\right)^2+\left(y^2\right)^2+2x^3y+2xy^3+2x^2y^2\right]\)
\(=2\left[x^4+x^2y^2+y^4+2x^3y+2xy^3+2x^2y^2\right]\)
\(=2x^4+2x^2y^2+2y^4+4x^3y+4xy^3+4x^2y^2\)
\(=x^4+y^4+\left(x^4+4x^3y+6x^2y^2+4xy^3+y^2\right)\)
\(=x^4+y^4+\left(x+y\right)^4=VP\)
Vậy \(x^4+y^4+\left(x+y\right)^4=2\left(x^2+xy+y^2\right)^2\) (đpcm)
Chúc bạn học tốt!!!
Thằng hiếu đã đánh tan vế trái thì anh đây đánh tan vế trái
\(VT=x^4+y^4+\left(x+y\right)^4\)
\(=\left(x^2+y^2\right)^2-2\left(xy\right)^2+\left(x+y\right)^4\)
\(=\left[\left(x+y\right)^2-2xy\right]^2-2\left(xy\right)^2+\left(x+y\right)^4\)
\(=\left(x+y\right)^4-4xy\left(x+y\right)^2+\left(2xy\right)^2-2\left(xy\right)^2+\left(x+y\right)^4\)
\(=2\left[\left(x+y\right)^4-4xy\left(x+y\right)^2+x^2y^2\right]\)
\(=2\left[\left(x+y\right)^2-xy\right]^2\)
\(=2\left(x^2+xy+y^2\right)^2=VP\)
