Question

Phân tích đa thức thành nhân tử

1. x⁴ + 6x³ +7x² - 6x +1

2. x⁴ + 2x³ +x² +x +1

3. (a-b)c³ - (a-c)b³ +(b-c)a³

4. a³(b² - c²) + b³(c²- a²)+ c³(a² - b²)

Answer 1

1.

\(x^4+6x^3+7x^2-6x+1\)

\(=(x^4+6x^3+9x^2)-2x^2-6x+1\)

\(=(x^2+3x)^2-2(x^2+3x)+1\)

\(=(x^2+3x-1)^2\)

2.

\(x^4+2x^3+x^2+x+1\)

\(=(x^4+2x^3+x^2)+(x+1)\)

\(=(x^2+x)^2+(x+1)=x^2(x+1)^2+(x+1)\)

\(=(x+1)[x^2(x+1)+1]=(x+1)(x^3+x^2+1)\)

Answer 2

3.

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

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

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

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

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

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

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

4.

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

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

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

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

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

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

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

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

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

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