by Guest » Fri Mar 01, 2019 3:52 pm
You can simplify this a little by letting y= x+ 4 so that x+ 2= y- 2 and then [tex](x+ 2)^4+ (x+ 4)^4= (y- 2)^4+ y^4=[/tex][tex]y^4- 8y^3+ 24y^2- 32x+ 16+ y^4= 2y^4- 8y^3+ 24y^2- 32y+ 16= 82[/tex]. Subtracting 82 from both sides, [tex]2y^4- 8y^3+ 24y^2- 32y+ 66= 0[/tex]. Divide by 2: [tex]y^4- 4y^3+ 12y^2- 16y+ 33= 0[/tex]. By the "rational root theorem" the only possible rational roots are 1, -1, 33, and -33. Evaluating the polynomial at those values shows that -33 is the only rational number root.