Chứng minh hằng đẳng thức sau: x^4 +y^4 +(x+y)^4

* Hướng dẫn giải

$\begin{array}{l}VT=x^4+y^4+{\left(x+y\right)}^4\\ =x^4+y^4+{\left(x^2+y^2+2xy\right)}^2\\ =2x^2{y}^2+\left[{\left(x^2+y^2\right)}^2+4xy\left(x^2+y^2\right)+4x^2{y}^2\right]\\ =x^4+y^4+\left(x^4+y^4+4x^2{y}^3+4x^3{y}^2+6x^2{y}^2\right)\\ =2x^4+2y^4+4x^2{y}^3+4x^3{y}^2+4x^2{y}^2+6x^2{y}^2\\ =2\left(x^4+y^4+2x^2{y}^3+2x^3{y}^2+3x^2{y}^2\right)\\ =2\left[\left(x^4+y^4+2x^2{y}^2\right)+2xy\left(x^2+y^2\right)+x^2{y}^2\right]\\ =2{\left(x^2+y^2+xy\right)}^2\\ =VP\\ \Rightarrow VT=VP\end{array}$