Question

证明：对任何整数x、y,整式x5 +3x4y- 5x3y2 -15x2y3+4xy4+12y5的值都不会等于2018 Q

Answer 1

x^5 +3x^4y- 5x^3y^2 -15x^2y^3+4xy^4+12y^5=x^4 (x+3y)- 5x^2y^2(x+3y) +4y^4(x+3y)

=(x^4- 5x^2y^2+4y^4)(x+3y)=(x^2-y^2)(x^2-4y^2)(x+3y)=(x+3y)(x+2y)(x-2y)(x+y)(x-y),

显然这五个因不可能互相相同的,不然x=y=0,原式为0了；而2018=2*1009,1009是质数,所以2018不可能拆成五个不相同的数相乘.所以原式不可能值为2018.