If P(x)=$a_0x^7$ + $a_1x^6$ +...+$a_6x +a_7$ has 7 integers values of x for which P(x)=1 OR P(x)=-1 then prove that P(x) is product of two polynomials

Question

Those two polynomials have integer coefficients. I tried to look at this as system of 7 equations and 7 variables. $a_0x1^7 + $a_1x1^6...+$a_6x1=$1-$a_7$ Or $a_0x1^7 + $a_1x1^6...+$a_6x1=-$(1+$a_7)$ We get that $a_6=$1-$a_7 -$($a_0x1^6 + $a_1x1^5...+$a_5x1)$ Therefore 1-$a_7 $ have to be divisible by x. So $a_7=1-k*x$ or $a_7=-1-k*x$