*Problem Solving through Problems,*and I am now on to chapter 4 in the section on polynomials. A few problems I have encountered so far involve polynomials of the form:

*P(x) = 1 + x + x*

^{2}+ x^{3}+....+x^{n}I noticed that when

*n*, the degree of the polynomial, is a sum of the first

*k*powers of two (starting from 0), then

*P(x)*can be factored as such:

*P(x) = (x+1)(x*

^{2}+1)(x^{4}+ 1)....(x^{2k}+ 1)For example, 7 = 2

^{0}+ 2

^{1}+ 2

^{2}so:

*1 + x*

^{2}+ x^{3}+ .... + x^{7 }= (x + 1)(x^{2}+1)(x^{4}+ 1)Furthermore, if the powers in the polynomial are successive multiples of

*m*(starting from 0) then the powers in the factorization are the first

*k*powers of two times

*m*, for example:

1 + x

1 + x

^{4}+ x^{8}+ x^{12}= (x^{4}+1)(x^{8}+1)Lastly, the same is true for an arithmetic progression of powers if the lowest power can be factored out to leave the appropriate form, for example:

*x*

^{2}+ x^{3}+ x^{4}+ x^{5}= x^{2}(*1 + x + x*^{2}+ x^{3}) = x^{2}*(x + 1)(x*

^{2}+1)A proof of this lemma would be immensely useful, because if it is true then it provides extremely clean solutions to a number of the problems in the chapter, but I'm at a loss as to how to prove it.