The sum of all the rational roots of the equation 6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 is

Step-by-step solution

Given, equation aligned & 6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 & 6 (x^6-1 )-25 x (x^4-1 )+31 x^2 (x^2-1 )=0 & (x^2-1 ) [6 (x^4+x^2+1 )-25 x (x^2+1 )+31 x^2 ]=0 & (x^2-1 ) [6 x^4-25 x^3+37 x^2-25 x+6 ]=0 aligned Either x^2-1=0 aligned & x=+1,-1 & or 6 x^4-25 x^3+37 x^2-25 x+6=0 & 6 (x^4+1 )-25 x (x^2+1 )+37 x^2=0 & 6 (x^2+ 1 x^2 )-25 (x+ 1 x )+37=0 & 6 (x+ 1 x )^2-25 (x+ 1 x )+25=0 & 6 (x+ 1 x )^2-15 (x+ 1 x )-10 (x+ 1 x )+25=0 & 3 (x+ 1 x ) [2 (x+ 1 x )-5 ]-5 [2 (x+ 1 x )-5 ]=0 & . 2 (x+ 1 x )-5 ] [3 (x+ 1 x )-5 ]=0