Question

一道因式分解题目，高手来啊：X^3+6X^2+11X+6（要求多种解）

道因式分解题目，高手来啊：X^3+6X^2+11X+6（要求多种解）
越多越好！！！哪个人最多分数谁拿！！！！！

Answer 1

法1：
X^3+6X^2+11X+6
=(X^3+X^2)+(5X^2+5X)+(6X+6)
=X^2(X+1)+5X(X+1)+6(X+1)
=(X+1)(X^2+5X+6)
=(X+1)(X+2)(X+3)

法2：
X^3+6X^2+11X+6
=(X^3+2X^2)+(4X^2+8X)+(3X+6)
=X^2(X+2)+4X(X+2)+3(X+2)
=(X+2)(X^2+4X+3)
=(X+2)(X+1)(X+3)

法3：
X^3+6X^2+11X+6
=(X^3+3X^2)+(3X^2+9X)+(2X+6)
=X^2(X+3)+3X(X+3)+2(X+3)
=(X+3)(X^2+3X+2)
=(X+3)(X+2)(X+1)

法4：
x^3+6x^2+11x+6
=(x^3+3x^2)+(3x^2+11x+6)
=x^2(x+3)+(x+3)(3x+2)
=(x+3)(x^2+3x+2)
=(x+3)(x+2)(x+1)

法5：
X^3+6X^2+11X+6
=x(x^2+6x+9)+2(x+3)
=x(x+3)^2+2(x+3)
=(x+3)(x^2+3x+2)
=(x+3)(x+1)(x+2)

法6：
设X^3+6X^2+11X+6＝(X+a)(X+b)(X+c)
则X^3+6X^2+11X+6＝X^3+(a+b+c)X^2+(ab+bc+ca)X+abc,
对比等式两边的系数，得
a+b+c=6,ab+bc+ca=11,abc=6
解方程组得a=1,b=2,c=3
所以X^3+6X^2+11X+6＝(X+1)(X+2)(X+3)

Answer 2

(x^3+6x^2+11x+6)
=(x^3+3x^2)+(3x^2+11x+6)
=x^2(x+3)+(x+3)(3x+2)
=(x+3)(x^2+3x+2)
=(x+3)(x+2)(x+1)

另外可以用待定系数法

Answer 3

拆项法：1)X^3+6X^2+11X+6
=x(x^2+6x+9)+2(x+3)
=x(x+3)^2+2(x+3)
=(x+3)(x^2+3x+2)
=(x+3)(x+1)(x+2)
2)(x^3+6x^2+11x+6)
=(3x^2+11x+6) +(x^3+3x^2)+
=x^2(x+3)+(x+3)(3x+2)
=(x+3)(x^2+3x+2)
=(x+1)(x+2)(x+3)

Answer 4

(x^3+6x^2+11x+6)

=(x^3+3x^2)+(3x^2+11x+6)

=x^2(x+3)+(x+3)(3x+2)

=(x+3)(x^2+3x+2)

=(x+3)(x+2)(x+1)

Answer 5

x^3+6x^2+11x+6=x<x^2+6x+11>+6=x<x^2+6x+5>+6x+6
=x<x+2><x+3>+6<x+1>