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\(x\left(y^2-z^2\right)+z\left(x^2-y^2\right)+y\left(z^2-x^2\right)\)
\(=x\left(y^2-z^2\right)-\left(y^2-z^2+z^2-x^2\right)z+y\left(z^2-x^2\right)\)
\(=x\left(y^2-z^2\right)-z\left(y^2-z^2\right)-z\left(z^2-x^2\right)+y\left(z^2-x^2\right)\)
\(=\left(y^2-z^2\right)\left(x-z\right)+\left(z^2-x^2\right)\left(y-z\right)\)
\(=\left(y-z\right)\left(z-x\right)\left(-\left(y+z\right)+z+x\right)\)
= \(\left(y-z\right)\left(z-x\right)\left(x-y\right)\)
\(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2+xy+yz+zx\right)\)
Lời giải:
a)
\(x^6+2x^5+x^4-2x^3-2x^2+1\)
\(=(x^6+2x^5+x^4)-2(x^3+x^2)+1\)
\(=(x^3+x^2)^2-2(x^3+x^2)+1=(x^3+x^2-1)^2\)
b)
\([4abcd+(a^2+b^2)(c^2+d^2)]^2-4[cd(a^2+b^2)+ab(c^2+d^2)]^2\)
\(=[4abcd+(a^2+b^2)(c^2+d^2)]^2-[2cd(a^2+b^2)+2ab(c^2+d^2)]^2\)
\(=[4abcd+(a^2+b^2)(c^2+d^2)+2cd(a^2+b^2)+2ab(c^2+d^2)][4abcd+(a^2+b^2)(c^2+d^2)-2cd(a^2+b^2)-2ab(c^2+d^2)]\)
\(=[(a^2+b^2)(c^2+d^2+2cd)+2ab(c^2+d^2+2cd)][(a^2+b^2)(c^2+d^2-2cd)-2ab(c^2+d^2-2cd)]\)
\(=[(a^2+b^2)(c+d)^2+2ab(c+d)^2][(a^2+b^2)(c-d)^2-2ab(c-d)^2]\)
\(=(c+d)^2(a^2+b^2+2ab)(c-d)^2(a^2+b^2-2ab)\)
\(=(c+d)^2(a+b)^2(c-d)^2(a-b)^2\)