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Let f(x), g(x) $\in \mathbb C[x].$
Prove that if f(x) | g(x) and g(x) | f(x), then there exists a nonzero
$c \in \mathbb C$ such that $f(x) = c * g(x)$
(You may use the fact that for any p(x), q(x) $\in \mathbb C [x],$
deg(p(x)q(x)) = deg(p(x)) + deg(q(x)).)

How would i begin the proof for this question?

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    $\begingroup$ Start by writing down what it means that $f$ divides $g$. And that $g$ divides $f$. Then use the degree formula. $\endgroup$
    – Daniel Fischer
    Mar 29, 2014 at 23:49

2 Answers 2

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Hint $\ $ Let $\ h=g/f.\,$ Then $\,h\,$ and $\,1/h\,$ are both polynomials so $\,\deg\,h = \,\ldots\,\Rightarrow\, h\in \Bbb C$

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    $\begingroup$ If you allow $\rm\,0\mid f\ (\!\iff\! f=0)\,$ then you need to also handle the degenerate cases when $\,f,g=0.\ \ $ $\endgroup$
    – Bill Dubuque
    Mar 30, 2014 at 0:29
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You'll find the following fact useful. Think about why it's true.

If $f(x)$ and $g(x)$ are (nonzero) polynomials with complex coefficients and $f(x)\mid g(x)$ then $\deg(f)\leq\deg(g)$.

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