Let $F$ be a field and let $a \neq 0$ be a zero of the polynomial $a_0 + a_1x + . . . +a_nx^n$ in $F[x]$.
I want to show that $\frac{1}{a}$ is a zero of the polynomial $a_n + a_{n-1}x + . . . + a_0x^n$
How can I do this?
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1Hint: try $y=\frac 1x$. – vadim123 Feb 27 '14 at 03:50
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Hint $ $ The second reversed (aka reciprocal) polynomial is simply $\ \hat f(x)= x^n f(1/x),\,\ n = \deg f.\,$ Now verify that $\hat f(1/a) = 0\,$ since $\,f(a) = 0.$
Bill Dubuque
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It just seems like the problem wants me to use the originally polynomial to derive that the reversed one has 1/a as a root. – terrible at math Feb 27 '14 at 04:02
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@ter That's precisely what you'll be doing if you follow my hint, since the proof that $,\hat f(1/a) = 0,$ uses $,f(a) = 0.$ – Bill Dubuque Feb 27 '14 at 04:04