Simple question regarding polynomials

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If k is a polynomial in x and y such that k(x-1) equals some polynomial q in y, then k must be zero. The reasoning is that for k to yield a polynomial in y when multiplied by (x-1), k cannot have any terms dependent on x, which would prevent q from being a polynomial solely in y. If k were not zero, it would introduce x-dependent terms that cannot be eliminated. Thus, the only solution that satisfies the condition is k = 0. This confirms that k must indeed equal zero for the equation to hold true.
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Hello all I had a simple question that I am intuitively sure I know the answer to but can't quite prove it.

Suppose k is a polynomial in x and y, and k(x-1) = q for q some polynomial in y. Then is k = 0 ?

How do I verify that k must be equal to 0? I can see that to just get a polynomial in y we have to try to get rid of that x term, but I can't quite prove why we can't just make some polynomial that gets rid of it somehow.

any help would be appreciated, thanks
 
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Suppose k != 0, so q = ...
 
If k=0, then k(x-1)=q=0.

If you let a polynomial in y be P(y) then if k=P(y)/(x-1), q=P(y)...
 

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