已知:a、b满足a3-3a2+5a=l,b3-3b2+5b=5,则a+b=________•
2
分析:对a
3-3a
2+5a=1,b
3-3b
2+5b=5分别进行整理后,将(a-1)与(b-1)看成整体,再化简求出(a-1)与(b-1)之间关系,即可求出a+b的值.
解答:∵a
3-3a
2+5a=1,
∴(a-1)
3+2(a-1)+2=0,
∵b
3-3b
2+5b=5,
∴(b-1)
3+2(b-1)-2=0,
设a-1=x,b-1=y,
则x
3+2x+2=0,y
3+2y-2=0,
两式相加可得x
3+y
3+2(x+y)=0,
化简整理得(x+y)(
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/241551.png)
![](http://thumb.1010pic.com/pic5/latex/241552.png)
)=0,
∴x+y=0,
即a-1+b-1=0,
∴a+b=2.
故答案为:2.
点评:本题考查因式分解的运用,解答此类题目时,要先考虑提取公因式,注意运用整体代入法求解.