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\(x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)\)
\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2\)
\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2-xyz+xyz\)
\(=\left(yz^2-xz^2-xyz+x^2z\right)-\left(zy^2-xyz-xy^2+x^2y\right)\)
\(=z\left(yz-xz-xy+x^2\right)-y\left(zy-xz-xy+x^2\right)\)
\(=\left(z-y\right)\left(yz-xz-xy+x^2\right)\)
\(=\left(z-y\right)\left[y\left(z-x\right)-x\left(z-x\right)\right]\)
\(=\left(z-y\right)\left(y-x\right)\left(z-x\right)\)
a)\(x^4-y^4=\left(x^2-y^2\right)\left(x^2+y^2\right)=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\)
b)\(x^2+11x+24=x^2+3x+8x+24=x\left(x+3\right)+8\left(x+3\right)=\left(x+3\right)\left(x+8\right)\)
c)\(xy\left(x-y\right)+y^2\left(y-z\right)+zx\left(z-x\right)=x\left[y\left(x-y\right)+z\left(z-x\right)\right]+y^2\left(y-z\right)\)
\(=x\left(xy-y^2+z^2-xz\right)+y^2\left(y-z\right)\)\(=x\left[\left(xy-xz\right)-\left(y^2-z^2\right)\right]+y^2\left(y-z\right)\)
\(=x\left[x\left(y-z\right)-\left(y-z\right)\left(y+z\right)\right]+y^2\left(y-z\right)\)\(=x\left(y-z\right)\left(x-y-z\right)+y^2\left(y-z\right)\)
\(=\left(y-z\right)\left(x^2-xy-xz\right)+y^2\left(y-z\right)=\left(y-z\right)\left(x^2-xy-xz+y^2\right)\)
a) xy(x + y) + yz(y + z) + xz(z + x) + 3xyz
= xy(X + y + z) + yz(x + y + z) + xz(X + y + z)
= (x + y +z)(xy + yz+ xz)
b) xy(x + y) - yz(y + z) - xz(z - x)
= x2y + xy2 - y2z - yz2 - xz2 + x2z
= x2(y + z) - yz(y + z) + x(y2 - z2)
= x2(y + z) - yz(y + z) + x(y + z)(y - z)
= (y + z)(x2 - yz + xy - xz)
= (y + z)[x(x + y) - z(x + y)]
= (y + z)(x + y)(x - z)
c) x(y2 - z2) + y(z2 - x2) + z(x2 - y2)
= x(y - z)(y + z) + yz2 - yx2 + x2z - y2z
= x(y - z)(y + z) - yz(y - z) - x2(y - z)
= (y - z)((xy + xz - yz - x2)
= (y - z)[x(y - x) - z(y - x)]
= (y - z)(x - z)(y -x)
a) \(\left(x+a\right)\left(x+2a\right)\left(x+3a\right)\left(x+4a\right)+a^4\)
\(=\left[\left(x+a\right)\left(x+4a\right)\right]\cdot\left[\left(x+2a\right)\left(x+3a\right)\right]+a^4\)
\(=\left(x^2+5ax+4a^2\right)\left(x^2+5ax+6a^2\right)+a^4\)
\(=\left(x^2+5ax+5a^2-a^2\right)\left(x^2+5ax+5a^2+a^2\right)+a^4\)\
\(=\left(x^2+5ax+5a^2\right)^2-a^4+a^4\)
\(=\left(x^2+5ax+5a^2\right)^2\)
b) Đặt \(a=x^2+y^2+z^2\); \(b=xy+yz+xz\)
\(\left(x^2+y^2+z^2\right)\left(x+y+z\right)^2+\left(xy+yz+zx\right)^2\)
\(=a\left(a+2b\right)+b^2\)
\(=a^2+2ab+b^2=\left(a+b\right)^2\)
\(=\left(x^2+y^2+z^2+xy+yz+zx\right)^2\)
a) \left(x+a\right)\left(x+2a\right)\left(x+3a\right)\left(x+4a\right)+a^4(x+a)(x+2a)(x+3a)(x+4a)+a4
=\left[\left(x+a\right)\left(x+4a\right)\right]\cdot\left[\left(x+2a\right)\left(x+3a\right)\right]+a^4=[(x+a)(x+4a)]⋅[(x+2a)(x+3a)]+a4
=\left(x^2+5ax+4a^2\right)\left(x^2+5ax+6a^2\right)+a^4=(x2+5ax+4a2)(x2+5ax+6a2)+a4
=\left(x^2+5ax+5a^2-a^2\right)\left(x^2+5ax+5a^2+a^2\right)+a^4=(x2+5ax+5a2−a2)(x2+5ax+5a2+a2)+a4\
=\left(x^2+5ax+5a^2\right)^2-a^4+a^4=(x2+5ax+5a2)2−a4+a4
=\left(x^2+5ax+5a^2\right)^2=(x2+5ax+5a2)2
b) Đặt a=x^2+y^2+z^2a=x2+y2+z2; b=xy+yz+xzb=xy+yz+xz
\left(x^2+y^2+z^2\right)\left(x+y+z\right)^2+\left(xy+yz+zx\right)^2(x2+y2+z2)(x+y+z)2+(xy+yz+zx)2
=a\left(a+2b\right)+b^2=a(a+2b)+b2
=a^2+2ab+b^2=\left(a+b\right)^2=a2+2ab+b2=(a+b)2
=\left(x^2+y^2+z^2+xy+yz+zx\right)^2=(x2+y2+z2+xy+yz+zx)2
a,Từ giả thiết ta có
(x2+y2+z2)(x+y+z)2+(xy+yz+zx)2
=(x2+y2+z2)(x2+y2+z2+2xy+2yz+2zx)+(xy+yz+zx)2
Đặt x2+y2+z2=a
xy+yz+zx=b
=>(x2+y2+z2)(x2+y2+z2+2xy+2yz+2zx)+(xy+yz+zx)2
=a(a+2b)+b2
=a2+2ab+b2
=(a+b)2
=(x2+y2+z2+xy+yz+zx)2
câu b hơi dài mình gửi sau nhé
Ta có: 2(x^4+y^4+z^4)-(x^2+y^2+z^2)^2-2(x^2+y^2+z^2)(x+y+z)^2+(x+y+z)^4
Gọi x^4+y^4+z^4=a
x^2+y^2+z^2=b
x+y+z=c
=>2(x^4+y^4+z^4)-(x^2+y^2+z^2)^2-2(x^2+y^2+z^2)(x+y+z)^2+(x+y+z)^4=2a-b^2-2bc^2+c^4
=2a-2b^2+b^2-2bc^2+c^4
=2(a-b^2)+(b+c^2)^2
Ta có
2(a-b2)=2[x^4+y^4+z^4-(x^2+y^2+z^2)2]
=2[x^4+y^4+z^4-x^4-y^4-z^4-2x2y2-2y2z2-2z2x2]
=2.(-2)(x2y2+y2z2+z2x2)
=-4(x2y2+y2z2+z2x2)
Lại có
(b+c^2)^2
=[(x^2+y^2+z^2)+(x+y+z)2]2
=[(x^2+y^2+z^2)-(x^2+y^2+z^2)-2(xy+yz+zx)]2
=4(xy+yz+zx)2
=>2(a-b^2)+(b+c^2)^2
=-4(x2y2+y2z2+z2x2)+4(xy+yz+zx)2
=8xyz(x+y+z)