i ) \(\left(b-c\right)^3+\left(c-a\right)^3+\left(a-b\right)^3\)
\(=b^3-3b^2c+3bc^2-c^3+c^3-3c^2a+3ca^2-a^3+a^3-3a^2b+3ab^2-b^3\)
\(=-3b^2c+3bc^2-3c^2a+3ca^2-3a^2b+3b^2a\)
\(=3bc\left(c-b\right)-3a\left(c^2-b^2\right)+3a^2\left(c-b\right)\)
\(=\left(c-b\right)\left[3bc-3a\left(c+b\right)+3a^2\right]\)
\(=\left(c-b\right)\left(3bc-3ac-3ab+3a^2\right)\)
\(=\left(c-b\right)\left[3c\left(b-a\right)-3a\left(b-a\right)\right]\)
\(=3\left(c-b\right)\left(c-a\right)\left(b-a\right)\)
k ) Đặt \(x^2=a;y^2=b;z^2=c\) , ta có :
\(\left(a+b\right)^3+\left(c-a\right)^3-\left(b+c\right)^3\)
\(=a^3+b^3+3a^2b+3b^2a+c^3-3c^2a+3ca^2-a^3-b^3-c^3-3b^2c-3cb^2\)
\(=3a^2b+3b^2a-3c^2a+3ca^2-3b^2c-3cb^2\)
\(=3\left(a^2b+b^2a-c^2a+ca^2-b^2c-cb^2\right)\)
\(=3\left[a^2\left(b+c\right)+\left(b^2-c^2\right)a-bc\left(b+c\right)\right]\)
\(=3\left(b+c\right)\left[a^2+\left(b-c\right)a-bc\right]\)
\(=3\left(b+c\right)\left(a^2+ab-ac-bc\right)\)
\(=3\left(b+c\right)\left[a\left(a+b\right)-c\left(a+b\right)\right]\)
\(=3\left(b+c\right)\left(a-c\right)\left(a+b\right)\)
\(=3\left(y^2+z^2\right)\left(x^2-z^2\right)\left(x^2+y^2\right)\)
\(=3\left(y^2+z^2\right)\left(x-z\right)\left(x+z\right)\left(x^2+y^2\right)\)
i)
\((b-c)^3+(c-a)^3+(a-b)^3\)
\(=(b-c)^3+(c-a)^3+3(b-c)^2(c-a)+3(b-c)(c-a)^2-3(b-c)^2(c-a)-3(b-c)(c-a)^2+(a-b)^3\)
\(=[(b-c)+(c-a)]^3-3(b-c)^2(c-a)-3(b-c)(c-a)^2+(a-b)^3\)
\(=(b-a)^3-3(b-c)(c-a)[(b-c)+(c-a)]+(a-b)^3\)
\(=-3(b-c)(c-a)(b-a)=3(a-b)(b-c)(c-a)\)
k)
\((x^2+y^2)^3+(z^2-x^2)^3-(y^2+z^2)^3\)
\(=(x^2+y^2)^3+(z^2-x^2)^3+3(x^2+y^2)^2(z^2-x^2)+3(x^2+y^2)(z^2-x^2)^2-3(x^2+y^2)^2(z^2-x^2)-3(x^2+y^2)(z^2-x^2)^2-(y^2+z^2)^3\)
\(=[(x^2+y^2)+(z^2-x^2)]^3-3(x^2+y^2)^2(z^2-x^2)-3(x^2+y^2)(z^2-x^2)^2-(y^2+z^2)^3\)
\(=(y^2+z^2)^3-3(x^2+y^2)(z^2-x^2)[(x^2+y^2)+(z^2-x^2)]-(y^2+z^2)^3\)
\(=-3(x^2+y^2)(z^2-x^2)(y^2+z^2)\)
\(=3(x^2+y^2)(x^2-z^2)(y^2+z^2)\)
