若a+b=2,求a的3次方+b的3次方+6ab的值

(a+b)^2=a^2+2ab+b^2=4
a^2+b^2=4-2ab

a^3+b^3+6ab
=(a+b)(a^2-ab+b^2)+6ab
=2(4-2ab-ab)+6ab
=2(4-3ab)+6ab
=8

由a+b=2，两边乘以3ab，得
3a^2b＋3ab^2＝6ab
所以
a^3+b^3+6ab
＝a^3+3a^2b＋3ab^2＋b^3
＝（a＋b）^3
＝8

（a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)
∵a+b=2
∴代入上式得
2^3=a^3+b^3+3ab*2
8=a^3+b^3+6ab
a^3+b^3+6ab=8

a^3+b^3+6ab=a^3+3ab^2+3ba^2+b^3+3ab-3ab^2+3ab-3ba^2
=(a+b)^3+3ab(1-b)+3ab(1-a)
=8+3ab(1-+1-a)=8

(a+b)^3=a^3+b^3+3ab(a+b)=a^3+b^3+6ab=2^3=8

8