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(B) Given that $P(x)$ is a non-zero polynomial satisfying $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Differentiating both sides with respect to $x

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$2$

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(B) Given that $P(x)$ is a non-zero polynomial satisfying $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Differentiating both sides with respect to $x$,we get: $P'(1+x) = -P'(1-x)$. Setting $x=0$,we obtain: $P'(1) = -P'(1) \implies 2P'(1) = 0 \implies P'(1) = 0$. Since $P(1)=0$ and $P'(1)=0$,by the Factor Theorem,$(x-1)^2$ must be a factor of $P(x)$. To check if $m$ can be larger,consider $P(x) = (x-1)^2$. This satisfies $P(1+x) = (1+x-1)^2 = x^2$ and $P(1-x) = (1-x-1)^2 = (-x)^2 = x^2$. Thus,$P(1+x)=P(1-x)$ holds. Since $P(x) = (x-1)^2$ is a valid polynomial,the largest integer $m$ such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$ is $2$.

Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Let $m$ be the largest integer such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$. Then,$m$ equals

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