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 $\displaystyle P(a) = a^4 + Aa^3 + Ba^2 + Aa + 1 = 0$.
In order for $\displaystyle \frac{1}{a}$ to be defined, we cannot have $\displaystyle a=0$. Setting $\displaystyle a = 0$ we see that the above equation reduces to $\displaystyle 1 = 0$, so we can safely say that $\displaystyle a \neq 0$.
Notice that $\displaystyle P\left(\frac{1}{a}\right) = \left(\frac{1}{a}\right)^4 + A\left(\frac{1}{a}\right)^3 + B\left(\frac{1}{a}\right)^2 + A\left(\frac{1}{a}\right) + 1 = \frac{1}{a^4}+\frac{A}{a^3}+\frac{B}{a^2}+\frac{A} {a}+1$.
Multiply both sides by $\displaystyle a^4$ to remove fractions.
$\displaystyle a^4\cdot P\left(\frac{1}{a}\right)=a^4 + Aa^3 + Ba^2 + Aa + 1$
Look familiar?
I'm still shaky on the restriction $\displaystyle (2+B)^2 \neq 4A^2$, 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.