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b, \(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)
\(=\left(x-y\right)^2\left(x-y\right)-\left(y-z\right)^2\left[\left(x-y\right)+\left(z-x\right)\right]+\left(z-x\right)^2\left(z-x\right)\)
\(=\left(x-y\right)^2\left(x-y\right)-\left(y-z\right)^2\left(x-y\right)-\left(y-z\right)^2\left(z-x\right)+\left(z-x\right)^2\left(z-x\right)\)
\(=\left(x-y\right)\left[\left(x-y\right)^2-\left(y-z\right)^2\right]-\left(z-x\right)\left[\left(y-z\right)^2-\left(z-x\right)^2\right]\)
\(=\left(x-y\right)\left(x-y-y+z\right)\left(x-y+y-z\right)-\left(z-x\right)\left(y-z-z+x\right)\left(y-z+z-x\right)\)
\(=\left(x-y\right)\left(x-2y+z\right)\left(x-z\right)-\left(z-x\right)\left(y-2z+x\right)\left(y-x\right)\)
\(=\left(x-y\right)\left(x-2y+z\right)\left(x-z\right)-\left(x-z\right)\left(y-2z+x\right)\left(x-y\right)\)
\(=\left(x-y\right)\left(x-z\right)\left(x-2y+z-y+2z-x\right)\)
\(=\left(x-y\right)\left(x-z\right)\left(3z-3y\right)\)
\(=3\left(x-y\right)\left(x-z\right)\left(z-y\right)\)
c, \(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left[\left(y-x\right)-\left(z-x\right)\right]-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left(y-x\right)+y^2z^2\left(z-x\right)-z^2x^2\left(z-x\right)\)
\(=\left(x^2y^2-y^2z^2\right)\left(y-x\right)+\left(y^2z^2-z^2x^2\right)\left(z-x\right)\)
\(=y^2\left(x-z\right)\left(x+z\right)\left(y-x\right)+z^2\left(y-x\right)\left(x+y\right)\left(z-x\right)\)
\(=y^2\left(x-z\right)\left(x+z\right)\left(y-x\right)-z^2\left(y-x\right)\left(x+y\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(y-x\right)\left[y^2\left(x+z\right)-z^2\left(x+y\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left(y^2x+y^2z-z^2x-z^2y\right)\)
\(=\left(x-z\right)\left(y-x\right)\left[x\left(y^2-z^2\right)+yz\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left[x\left(y-z\right)\left(y+z\right)+yz\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left(y-z\right)\left(xy+xz+yz\right)\)
d, \(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3+z^3-3xyz-3xy\left(x+y\right)\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
a: \(=xy^2-xz^2+z^2y-x^2y+x^2z-zy^2\)
\(=-xy\left(x-y\right)-z^2\left(x-y\right)+z\left(x^2-y^2\right)\)
\(=\left(x-y\right)\left(-xy-z^2+zx+zy\right)\)
\(=\left(x-y\right)\left[xz-xy+zy-z^2\right]\)
\(=\left(x-y\right)\left[x\left(z-y\right)-z\left(z-y\right)\right]\)
\(=\left(x-y\right)\left(z-y\right)\left(x-z\right)\)
d:
Tham khảo:
1 , \(x^5+x^4+1=\left(x^5+x^4+x^3\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\)
= \(x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)=\(\left(x^2+x+1\right)\left(x^3-x+1\right)\)
2 , \(x\left(x+4\right)\left(x+6\right)\left(x+10\right)+128=\left(x^2+10x\right)\left(x^2+10x+24\right)+128\)(*)
Đặt x2 + 10 = a , a>0 (1)
=> (*) <=> a(a+24)+128=a2 + 24a+128=(a+8)(a+16) (**)
Thay (1) vào (**) ta được :
(*) <=> \(\left(x^2+10+8\right)\left(x^2+10+16\right)\)
a) x2 + xy - 2y2 = (x2 + xy + \(\dfrac{1}{4}\)y2) - \(\dfrac{9}{4}\)y2 = (x + \(\dfrac{1}{2}\)y)2 - (\(\dfrac{3}{2}\)y)2 = (x + \(\dfrac{1}{2}\)y - \(\dfrac{3}{2}\)y)(x + \(\dfrac{1}{2}\)y + \(\dfrac{3}{2}\)y) = (x - y)(x + 2y)
b) x5 + x + 1 = (x5 + x4 + x3) - (x4 + x3 + x2) + (x2 + x + 1) = x3(x2 + x + 1) - x2(x2 + x + 1) + (x2 + x + 1) = (x3 - x2 + 1)(x2 + x + 1)
Phân tích nốt cái x3 - x2 + 1 là xong. Đoạn này mình bấm máy 500MS không rõ nghiệm chính xác là bao nhiêu nên để dành cho bạn làm đó
a) x(\(y^2\)-\(z^2\))+y(\(z^2-z^2\)) + (\(x^2-y^2\))
=\(xy^2-xz^2+x^2z-y^2z\)
=\(y^2\left(x-z\right)+xz\left(x-z\right)\)
= \(y^2+xz\)
o: \(x^3-xy^2+x^2y-y^3\)
\(=\left(x-y\right)\left(x^2+xy+y^2\right)+xy\left(x-y\right)\)
\(=\left(x-y\right)\left(x^2+2xy+y^2\right)\)
\(=\left(x-y\right)\left(x+y\right)^2\)
p: \(a^3-ma-mb+b^3\)
\(=\left(a+b\right)\left(a^2-ab+b^2\right)-m\left(a+b\right)\)
\(=\left(a+b\right)\left(a^2-ab+b^2-m\right)\)
q: \(\left(3x+1\right)^3-\left(1-2x\right)^3\)
\(=\left(3x+1\right)^3+\left(2x-1\right)^3\)
\(=\left(3x+1+2x-1\right)\left[\left(3x+1\right)^2-\left(3x+1\right)\left(2x-1\right)+\left(2x-1\right)^2\right]\)
\(=5x\left[9x^2+6x+1-6x^2+3x-2x+1+4x^2-4x+1\right]\)
\(=5x\left(7x^2+5x+3\right)\)
a) \(2\left(x+y\right)^2-7\left(x+y\right)+5\)
\(=2\left(x+y\right)^2-2\left(x+y\right)-5\left(x+y\right)+5\)
\(=2\left(x+y\right)\left(x+y-1\right)-5\left(x+y-1\right)\)
\(=\left(x+y-1\right)\left(2x+2y-5\right)\)
b) c) tương tự
\(M=\left[x+\left(y-z\right)-2x\right]+y+z-\left(2-x-y\right)\)
\(=-x+y-z+y+z-2+x+y\)
\(=3y-2\)
\(N=x-\left[x-\left(y-z\right)-x\right]\)
\(=x-\left(-y+z\right)\)
\(=x+y-z\)
\(M+N=3y-2+x+y-z=x+4y-z-2\)
\(M-N=\left(3y-2\right)-\left(x+y-z\right)\)
\(=3y-2-x-y+z\)
\(=-x+2y+z-2\)
\(M=\left[x+\left(y-z\right)-2x\right]+y+z-\left(2-x-y\right)\\ M=x+y-z-2x+y+z-2+x+y\\ M=3y-2\)
\(N=x-\left[x-\left(y-z\right)-x\right]\\ N=x-\left(x-y+z-x\right)\\ N=x-x+y-z+x\\ N=x+y-z\)
\(M+N=3y-2+x+y-z\\ M+N=x+4y-z-2\)
\(M-N=3y-2-\left(x+y-z\right)\\ M-N=3y-2-x-y+z\\ M-N=-x+2y+z-2\)