Question

已知集合A={2,4,a³-2a²-a+7},B={-4,a+3,a²-2a+2,a³+a²+3a+7},若A∩B={2,5},求实数a的值,并求A∪B.

Answer 1

{-4,2,4,5,25}.

∵A∩B={2,5},∴5∈A,A={2,4,5}.
由已知可得a³-2a²-a+7=5.
∴a³-2a²-a+2=0.∴(a²-1)(a-2)=0.∴a=2或a=±1.
(1)当a=2时,B={-4,5,2,25},A∩B={2,5}与题设相符;
(2)当a=1时,B={-4,4,1,12},A∩B={4}与题设矛盾;
(3)当a=-1时,B={-4,2,5,4},A∩B={2,4,5}与题设矛盾.
综上(1)(2)(3)知a=2,且A∪B={2,4,5}∪{-4,5,2,25}={-4,2,4,5,25}.