Question

Phân tích đa thức sau thành nhân tử:

1)\(a\left(b^3-c^3\right)+b\left(c^3-a^3\right)+c\left(a^3-b^3\right)\)

2) \(a^3\left(b^2-c^2\right)+b^3\left(c^2-a^2\right)+c^3\left(a^2-b^2\right)\)

Answer 1

Câu 1:

\(a(b^3-c^3)+b(c^3-a^3)+c(a^3-b^3)\)

\(=a(b^3-c^3)-b(a^3-c^3)+c(a^3-b^3)\)

\(=a(b^3-c^3)-b[(b^3-c^3)+(a^3-b^3)]+c(a^3-b^3)\)

\(=(a-b)(b^3-c^3)-(b-c)(a^3-b^3)\)

\(=(a-b)(b-c)(b^2+bc+c^2)-(b-c)(a-b)(a^2+ab+b^2)\)

\(=(a-b)(b-c)(b^2+bc+c^2-a^2-ab-b^2)\)

\(=(a-b)(b-c)(c^2+bc-a^2-ab)\)

\(=(a-b)(b-c)[(c^2-a^2)+b(c-a)]\)

\(=(a-b)(b-c)(c-a)(c+a+b)\)

Answer 2

Câu 2:

\(a^3(b^2-c^2)+b^3(c^2-a^2)+c^3(a^2-b^2)\)

\(=a^3(b^2-c^2)-b^3(a^2-c^2)+c^3(a^2-b^2)\)

\(=a^3(b^2-c^2)-b^3[(b^2-c^2)+(a^2-b^2)]+c^3(a^2-b^2)\)

\(=(a^3-b^3)(b^2-c^2)-(b^3-c^3)(a^2-b^2)\)

\(=(a-b)(a^2+ab+b^2)(b-c)(b+c)-(b-c)(b^2+bc+c^2)(a-b)(a+b)\)

\(=(a-b)(b-c)[(b+c)(a^2+ab+b^2)-(a+b)(b^2+bc+c^2)]\)

\(=(a-b)(b-c)(a^2b+a^2c-ac^2-bc^2)\)

\(=(a-b)(b-c)[b(a^2-c^2)+ac(a-c)]\)

\(=(a-b)(b-c)[b(a-c)(a+c)+ac(a-c)]\)

\(=(a-b)(b-c)(a-c)(ab+bc+ac)\)