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.
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.