Let x=a be a root fo the quartic polynomial P(x) = x^4 + Ax^3 + Bx^2 + Ax + 1 where (2+B)^2=/ 4A^2
Show that x=1/a is a root
So we have .
In order for to be defined, we cannot have . Setting we see that the above equation reduces to , so we can safely say that .
Notice that .
Multiply both sides by to remove fractions.
I'm still shaky on the restriction , which seems unnecessary. I'll look into it but maybe someone with keener insight can elucidate. Or perhaps it's just unnecessary.
Ive pretty much got that. But the only thing i dont understand is that if we multiply a^4 to the polynomial, it changes dosent it. Since a^4 is not a roots of unity. Or are we to argue that since a^4 P(1/a) = 0 then p(1/a) = 0 via division?
The restriction of (2+B)^2 =/=4A^2 is i think unncessary. Think that is another part of the question.