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a) \(g\left(x,y\right)=x^2-10xy+9y^2=x^2-xy-9xy+9y^2\)
\(=x\left(x-y\right)-9y\left(x-y\right)=\left(x-y\right)\left(x-9y\right)\).
b )\(f\left(x,y\right)=x^6+x^4+x^2y^2+y^4-y^6\)
\(=x^6-y^6+x^4+x^2y^2+y^4\)
\(=\left(x^3\right)^2-\left(y^3\right)^2+\left(x^4+2x^2y^2+y^4\right)-x^2y^2\)
\(=\left(x^3-y^3\right)\left(x^3+y^3\right)+\left(x^2+y^2\right)^2-\left(xy\right)^2\)
\(=\left(x-y\right)\left(x^2+xy+y^2\right)\left(x+y\right)\left(x^2-xy+y^2\right)+\left(x^2+y^2-xy\right)\left(x^2+y^2+xy\right)\)
\(=\left(x^2+xy+y^2\right)\left(x^2-xy+y^2\right)\left[\left(x-y\right)\left(x+y\right)+1\right]\)
\(=\left(x^2+xy+y^2\right)\left(x^2-2y+y^2\right)\left(x^2-y^2+1\right)\)
Vậy \(f\left(x,y\right)=\left(x^2+xy+y^2\right)\left(x^2-xy+y^2\right)\left(x^2-y^2+1\right)\)
\(\left(2x+1\right)^2-\left(x-1\right)^2\)
\(\Leftrightarrow\left(2x+1-x+1\right)\left(2x+1+x-1\right)\)
\(\Leftrightarrow\left(x+2\right)3x\)
c) xét giá trị riêng
\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz\)
\(=xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+xyz+xyz\)
\(=xy\left(x+y\right)+y^2z+yz^2+x^2z+xz^2+xyz+xyz\)
\(=xy\left(x+y\right)+y^2z+xyz+yz^2+xz^2+x^2z+xyz\)
\(=xy\left(x+y\right)+yz\left(x+y\right)+z^2\left(x+y\right)+xz\left(x+y\right)\)
\(=\left(x+y\right)\left(xy+yz+z^2+xz\right)\)
\(=\left(x+y\right)\left[y\left(x+z\right)+z\left(x+z\right)\right]=\left(x+y\right)\left(y+z\right)\left(x+z\right)\)
a) \(x^2-y^2-x-y\)
\(=\left(x+y\right)\left(x-y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-1\right)\)
\(a,\left(2x+1\right)^2-\left(x-1\right)^2\\ =\left(2x+1-x+1\right)\left(2x+1+x-1\right)\\ =\left(x+2\right)3x\)
\(b,9\left(x+5\right)^2-\left(x-7\right)^2\\ =\left[3\left(x+5\right)\right]^2-\left(x-7\right)^2\\ =\left(3x+15-x+7\right)\left(3x+15+x-7\right)\\ =\left(2x+22\right)\left(4x+8\right)\)
\(c,x^2y+xy^2-x-y\\ =xy\left(x+y\right)-\left(x+y\right)\\ =\left(x+y\right)\left(xy-1\right)\)
Các câu sau tương tự
\(yz\left(y+z\right)+zx\left(z-x\right)-xy\left(x+y\right)\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left[\left(y+z\right)-\left(z-x\right)\right]\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left(y+z\right)+xy\left(z-x\right)\)
\(=y\left(y+z\right)\left(z-x\right)+x\left(z-x\right)\left(z-y\right)\)
\(=\left(z-x\right)\left(yz-xy+xz-xy\right)\)
Bài 1 :
\(e,x^2+2xy+y^2-2x-2y+1\)
\(=\left(x+y-1\right)^2\)
Bài 2:
\(b,2x^3+3x^2+2x+3=0\)
\(\Leftrightarrow\left(2x^3+2x\right)+\left(3x^2+3\right)=0\)
\(\Leftrightarrow2x\left(x^2+1\right)+3\left(x^2+1\right)=0\)
\(\Leftrightarrow\left(x^2+1\right)\left(2x+3\right)=0\)
\(\Leftrightarrow2x+3=0\left(x^2+1>0\right)\)
\(\Leftrightarrow x=-\dfrac{3}{2}\)
a) \(\left(x+y\right)^3-x^3-y^3\)
\(=\left(x+y\right)^3-\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(=\left(x+y\right)\left[\left(x+y\right)^2-x^2+xy-y^2\right]\)
\(=\left(x+y\right)\left(x^2+2xy+y^2-x^2+xy-y^2\right)\)
\(=3xy\left(x+y\right)\)
b) \(x^2+y^2+2xy+yz+xz\)
\(=\left(x^2+2xy+y^2\right)+\left(yz+xz\right)\)
\(=\left(x+y\right)^2+z\left(x+y\right)\)
\(=\left(x+y\right)\left(x+y+z\right)\)
c) \(x^2-10xy-1+25y^2\)
\(=\left(x^2-10xy+25y^2\right)-1\)
\(=\left(x-5y\right)^2-1\)
\(=\left(x-5y-1\right)\left(x-5y+1\right)\)
d) \(ax^2-ax+bx^2-bx+a+b\)
\(=(ax^2+bx^2)-(ax+bx)+(a+b)\)
\(=x^2(a+b)-x(a+b)+(a+b)\)
\(=(a+b)(x^2-x+1)\)
e)\(x^2-2y+3xz+x-2y+3z\)
\(=(x^2+x)-(2xy+2y)+(3xz+3z)\)
\(=x(x+1)-2y(x-1)+3z(x+1)\)
\(=(x+1)(x-2y+3z)\)
f) \(xyz-xy-yz-xz+x+y+z-1\)
\(=(xyz-xy)-(yz-y)-(xz-x)+(z-1)\)
\(=xy(z-1)-y(z-1)-x(z-1)+(z-1)\)
\(=(z-1)(xy-y-x+1)\)
\(=(z-1)[y(x-1)-(x-1)]\)
\(=(z-1)(x-1)(y-1)\)
_Học tốt_