Ta có :
\(VT=\left(x+y\right)^4+x^4+y^4\)
\(=\left[\left(x+y\right)^2\right]^2+x^4+y^4\)
\(=\left(x^2+2xy+y^2\right)^2+x^4+y^4\)
\(=x^4+y^4+4x^2y^2+4x^3y+4xy^3+2x^2y^2+x^4+y^4\)
\(=2\left(x^4+y^4+2x^2y^2+2x^3y+2xy^3+x^2y^2\right)\)
\(=2\left(x^2+xy+y^2\right)^2=VP\)
\(\left(đpcm\right)\)
CMR:\(x^4+y^4+\left(x+y\right)^4=2.\left(x^2+xy+y^2\right)^2\)
a)\(\left(a+b+c\right)^2+a^2+b^2+c^2=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)
b)\(x^4+y^4+\left(x+y\right)^4=2\left(x^2+xy+y^2\right)^2\)
Chứng minh:
\(\left(x-y\right)\)\(\left(x^3+x^2y+xy^2+y^3\right)\)\(=x^4-y^4\)
\(X^4+y^4+\left(x+y\right)^4=2\left(x^2+xy+y^2\right)^2\)Chứng minh hàng đẳng thức
Rút gọn biểu thức:
a) \(A=\left(x-y\right)^3+\left(y+x\right)^3+\left(y-x\right)^3-3xy\left(x+y\right)\)
b) \(B=3x^2\left(x+1\right)\left(x-1\right)-\left(x^2-1\right)\left(x^4+x^2+1\right)+\left(x^2-1\right)^3\)
c) \(C=\left(x+y\right)\left(x^2-xy+y^2\right)+\left(x-y\right)\left(x^2+xy+y^2\right)-2x^3\)
d) \(D=\left(x+1\right)^3+\left(x-1\right)^3+x^3-3x\left(x+1\right)\left(x-1\right)\)
Chứng minh các đẳng thức:
a)\(\left(x-y\right).\left(x^3+x^2y+xy^2+y^3\right)=x^4-y^4\)
b)\(\left(a+b\right)^2-\left(a-b\right)^2=4ab\)
Bài 1: Cm giá trị của biểu thức sau không phụ thuộc vào biến x:
a) \(\left(2x+3\right).\left(4x^2-6x+9\right)-2\left(4x^3-1\right)\)
b)\(\left(x+3\right)^3+\left(x+9\right).\left(x^2+27\right)\)
c)\(\left(x+y\right).\left(x^2-xy+y^2\right)+\left(x-y\right).\left(x^2+xy+y^2\right)-2x^3\)
Bài 2: Tìm x biết:
a) \(\left(x+2\right)^2-9=0\)
b) \(\left(x+2\right)^2-x^2+4=0\)
c) \(\left(x-3\right)^2-4=0\)
d) \(x^2-2x=24\)
e) \(\left(2x-1\right)^2+\left(x+3\right)^2-5\left(x-7\right)\left(x+7\right)=0\)
Rút gọn biểu thức :
a) \(2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2\)
b) P=\(\left(5x-1\right)+2\left(1-5x\right)\left(4+5x\right)+\left(5x+4\right)^2\)
c) Q=\(\left(x-y\right)^3+\left(y+x\right)^3+\left(y-x\right)^3-3xy\left(x+y\right)\)
d) P = \(12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
Cho x , y thỏa mãn \(\left\{{}\begin{matrix}x^2+y^2-xy=2\\x^4+y^4+4^2y^2=8\end{matrix}\right.\)
