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Just to make sure, are the following correct?

$\sum_{i=0}^{n} \alpha x_{i}=\alpha \sum_{i=0}^{n} x_{i}$

$\prod_{i=0}^{n} \alpha x_{i} =\alpha^{n}\prod_{i=0}^{n}x_{i}$

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    Except that the second one must have $\alpha^{n+1}$ – 2017-01-27
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    Check by expanding with $n=3$. – 2017-01-27
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    @OpenBall because the counter is from $0$ right? if it was from $1$ so it is $\alpha^{n}$? – 2017-01-27
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    Yes ${}{}{}{}{}$ – 2017-01-27

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Almost. First one is fine. On the second one you have the right idea but it should be $\alpha^{n+1}$ on the RHS, because when $i$ ranges from $0$ to $n$ it ($i$) takes $n+1$ total values.