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Sniffnoy comments on Unknown knowns: Why did you choose to be monogamous? - Less Wrong
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General fact: Summing a polynomial of degree k results in a polynomial of degree k+1. This is easier to see if you use the x choose k basis, rather than the x^k basis.
This is even easier to see if you remember that summation is discrete integration, which is the opposite of differentiation, which reduces degree by one. I recommend Graham, Knuth and Patashnik's "Concrete Mathematics" for stuff like this.