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A known-plaintext attack on an LFSR-based stream cipher

I am reading a book about Cryptography, and I am stuck in a question. I am really trying to solve it for weeks. But I think the problem is I couldn't understand the whole picture. The question was ...
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How to prove this LFSR equation?

I am having difficulty solving part (ii) of Excercise 5.17 from Becker's Cipher Systems. Exercise 5.17 $\quad$If $f(x),g(x)$ are any two polynomials over $GF(2)$ with $f(0)=g(0)=1$ show that ...