MHB Prove Product of Polynomials: No Odd Degree Terms

[anemone]

Prove that in the following product

$P=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$

after multiplying and collecting like terms, there does not appear a term in $x$ of odd degree.

 

[kaliprasad]

Spoiler: solution

I call it P(x) a polynomial

$P(x) =(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$

So $(1-x^2)P(x) =(1-x)(1+x) P(x) = (1+x) (1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1-x) (1+x+x^2+x^3+\cdots+x^{99}+x^{100})$
OR
$(1-x^2)P(x) =(1+x^{101})(1-x^{101})=(1-x^{202})$

OR
$P(x) =\frac{1-x^{202}}{1-x^2} = (1+x^2 + x^4+x^6+\cdots+x^{198}+x^{200})$

So no x term with odd degree