Bài 4:
\(x^6-2x^4-x^3y^3+2xy^3\)
\(=\left(x^6-2x^4\right)-\left(x^3y^3-2xy^3\right)\)
\(=x^4\left(x^2-2\right)-xy^3\left(x^2-2\right)\)
\(=\left(x^2-2\right)\left(x^4-xy^3\right)\)
\(=x\left(x^2-2\right)\left(x^3-y^3\right)\)
\(=x\left(x^2-2\right)\left(x-y\right)\left(x^2+xy+y^2\right)\)
Bài 5:
\(\left(a+b+c\right)^2+\left(a-b+c\right)^2-4b^2\)
\(=2a^2+2b^2+2c^2+2ab+2ac+2bc-2ab-2bc+2ac-4b^2\)
\(=2a^2+2b^2+2c^2+4ac-4b^2\)
\(=2a^2-2b^2+2c^2+4ac\)
\(=2\left[\left(a^2+2ac+c^2\right)-b^2\right]\)
\(=2\left[\left(a+c\right)^2-b^2\right]\)
\(=2\left(a+b+c\right)\left(a-b+c\right)\)
Bài 6:
\(a\left(b^2-c^2\right)-b\left(c^2-a^2\right)+c\left(a^2-b^2\right)\)
\(=ab^2-ac^2-bc^2+a^2b+ca^2-b^2c\)
\(=\left(ab^2-b^2c\right)-\left(ac^2-ca^2\right)+\left(a^2b-bc^2\right)\)
\(=b^2\left(a-c\right)-ac\left(c-a\right)+b\left(a^2-c^2\right)\)
\(=b^2\left(a-c\right)+ac\left(a-c\right)+b\left(a-c\right)\left(a+c\right)\)
\(=\left(a-c\right)\left[b^2+ac+b\left(a+c\right)\right]\)
\(=\left(a-c\right)\left(b^2+ac+ab+bc\right)\)
\(=\left(a-c\right)\left[\left(b^2+ab\right)+\left(ac+bc\right)\right]\)
\(=\left(a-c\right)\left[b\left(a+b\right)+c\left(a+b\right)\right]\)
\(=\left(a-c\right)\left(a+b\right)\left(b+c\right)\)
