因式分解——疑难杂症大集合1.16XXXX-72XXYY+81YYYY-8XX-18YY+12.XXX+(2A+1)XX+(AA+2A-1）X+AA-13.2X^5+(6+3Y)X^4-(2-9Y+YY)XXX-3(2+Y+2YY)XX-(9Y-2YY)XX-(9Y-2YY)X+6YY4.2^67-12^67-1=1+2+4+……+2^66我当然知道，但你还没有分解呢

Question

因式分解——疑难杂症大集合
1.16XXXX-72XXYY+81YYYY-8XX-18YY+1
2.XXX+(2A+1)XX+(AA+2A-1）X+AA-1
3.2X^5+(6+3Y)X^4-(2-9Y+YY)XXX-3(2+Y+2YY)XX-(9Y-2YY)XX-(9Y-2YY)X+6YY
4.2^67-1
2^67-1=1+2+4+……+2^66
我当然知道，但你还没有分解呢

Answer 1

1.16XXXX-72XXYY+81YYYY-8XX-18YY+1
=(4xx-9yy)^2+2(4xx-9yy)+1-16xx
=(4xx-9yy+1)^2-(4x)^2
=(4xx-9yy-4x+1)(4xx-9yy+4x+1)
2.XXX+(2A+1)XX+(AA+2A-1）X+AA-1
=(X+1)*XX+2AX(X+1)+AA*(X+1)-(X+1)
=(X+1)(XX+2AX+AA-1)
=(X+1)[(X+A)^2-1^2]
=(X+1)(X+A-1)(X+A+1)
4.2^67-1
由m^n-1=(m-1)[m^(n-1)+m^(n-2)+m^(n-3)……+1]
可得2^67-1=1+2+4+……+2^66