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a, Xét tử thức \(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left[\left(x-z\right)-\left(y-z\right)\right]\)
\(=x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-z\right)-z^2\left(y-z\right)\)
\(=\left(x^2-z^2\right)\left(y-z\right)-\left(y^2-z^2\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(x+z\right)\left(y-z\right)-\left(y-z\right)\left(y+z\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(xy-xz+yz-z^2-y^2-yz+yz+z^2\right)\)
\(=\left(x-z\right)\left(xy-xz+yz-y^2\right)=\left(x-z\right)\left[x\left(y-z\right)-y\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(x-y\right)\left(y-z\right)\)
Mẫu thức \(x^2y-x^2z+y^2z-y^3=x^2\left(y-z\right)-y^2\left(y-z\right)=\left(x-y\right)\left(x+y\right)\left(y-z\right)\)
Vậy \(\frac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)}{x^2y-x^2z+y^2z-y^3}=\frac{\left(x-y\right)\left(y-z\right)\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}=\frac{x-z}{x+y}\)
b, \(\frac{x^5+x+1}{x^3+x^2+x}=\frac{x^5-x^2+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{x^2\left(x-1\right)\left(x^2+x+1\right)+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{\left(x^2+x+1\right)\left(x^3-x^2+1\right)}{x\left(x^2+x+1\right)}=\frac{x^3-x^2+1}{x}\)
b) Ta có nhận xét này nếu a+b+c=0 thì\(a^3+b^3+c^3=3abc\) (nếu cần chứng minh thì hỏi sau nhé)
Khi đó: tử=(x-y)(y-z)(z-x)
Mẫu nó cứ thế nào ấy. Rút gọn cũng chỉ được một chút thôi, chẳng gọn lắm
a) chịu chưa nghĩ ra
Dat (x-y)2+(y-z)2+(x-z)2=A
=(x+y)3+z3-3x2y-3xy2-3xyz / A
=(x+y+z).(x2+2xy+y2-xy-yz+z2)-3xy(x+y+z) / A
=(x+y+z).(x2+y2+z2-xy-yz-xz) /A
=2(x+y+z).(x2+y2+z2-xy-yz-xz) /2A
=(x+y+z)[ (x2-2xy+y2)+(y2-2yz+z2)+(x2-2xz+z2) / 2A
=(x+y+z).[ (x-y}2+(y-z)2+(x-z)2 ] /2A
=(x+y+z). A /2A
=x+y+z /2
a)\(9x^2+30x+25+9x^2-30x+25-\left(9x^2-2^2\right)\)
=\(9x^2+54\)=\(9\left(x^2+6\right)\)
b)\(2x\left(4x^2-4x+1\right)-3x\left(x^2-9\right)-4x\left(x^2+2x+1\right)\)
=\(8x^3-8x^2+2x-3x^3+27x-4x^3-8x^2-4x\)
=\(x^3-16x^2+25x\)
c)\(\left(x+y-z\right)^2-2\left(x+y-z\right)\left(x+y\right)+\left(x+y\right)^2\)
=\(\left(x+y-z-\left(x+y\right)\right)^2\)=\(\left(-z\right)^2\)