题目

几道因式分解题要求助!
x^4+4x^3-12x-9
x^3-y^3-x^2-xy-y^2
a^8+a^4b^4+b^8
(a^2-1)(b^2-1)-4ab
麻烦给出过程和思路

提问时间：2020-11-11

答案

x^4+4x³-12x-9,补回x²项
=x^4+4x³+3x²-3x²-12x-9
=x²(x²+4x+3)-3(x²+4x+3)
=(x²+4x+3)(x²-3)
=(x+1)(x+3)(x²-3)
=(x+1)(x+3)(x+√3)(x-√3)
x³-y³-x²-xy-y²,补回x²y项和xy²项
=x³+x²y+xy²-x²y-xy²-y³-x²-xy-y²
=x(x²+xy+y²)-y(x²+xy+y²)-(x²+xy+y²)
=(x-y-1)(x²+xy+y²)
a^8+(a^4)(b^4)+b^8,记得a²+b²=(a+b)²-2ab
=a^8+2(a^4)(b^4)+b^8-(a^4)(b^4)
=(a^4+b^4)²-(a²b²)²
=(a^4+b^4+a²b²)(a^4+b^4-a²b²)
=(a^4+2a²b²+b^4-a²b²)(a^4-a²b²+b^4)
=[(a²+b²)²-(ab)²](a^4-a²b²+b^4)
=(a²+b²-ab)(a²+b²+ab)(a^4-a²b²+b^4)
(a²-1)(b²-1)-4ab,拆开再重组
=a²b²-a²-b²-4a²+1
=a²b²-4ab-a²-b²+1
=a²b²-2ab+1-a²-2ab-b²
=a²b²-2ab+1-(a²+2ab+b²)
=(ab-1)²-(a+b)²
=[(ab-1)-(a+b)][(ab-1)+(a+b)]
=(ab-1-a-b)(ab-1+a+b)