Question

Phân tích đa thức sau thành nhân tử :

a, a³.(b - c) + b³.(c - a) + c³.(a - b)

b, a.(b² + c²) + b.(c² + a² ) + c.(a² + b²) + 2.a.b.c

c, a⁴.(b - c) + b⁴.(c - a) + c⁴.(a - b)

Answer 1

a)

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

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

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

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

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

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

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

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

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

Answer 2

b)

\(a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)+2abc\)

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

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

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

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

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

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

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

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

Answer 3

c)

\(a^4(b-c)+b^4(c-a)+c^4(a-b)\)

\(=a^4(b-c)-b^4[(b-c)+(a-b)]+c^4(a-b)\)

\(=a^4(b-c)-b^4(b-c)-b^4(a-b)+c^4(a-b)\)

\(=(a^4-b^4)(b-c)-(b^4-c^4)(a-b)\)

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

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

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

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

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

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

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