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Suppose $f(X) = (X − α)^r\cdot g(X)$, where $α ∈ \Bbb C$ is nonzero, $r ∈ \Bbb Z^+$, and $g ∈ \Bbb C[X]$ is nonzero. Prove that $$||g|| < (1 + \deg g)\cdot (2 \max(1, |α|^{−1}))\cdot \deg f\cdot ||f||$$

I wanted to try Newton's theory but failed. I am considering using substitution.

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What is your definition of $\| \cdot \|$? –  Robert Israel Nov 12 '12 at 8:43
    
∥⋅∥ means the highest coefficient of monomial within the polynomial. –  god jacky Nov 12 '12 at 13:37

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