For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of $\frac{\sin x}x=\prod_{n=1}^\infty(1-\frac{x^2}{n^2\pi^2})$ is unjustified without Weierstrass' factorization theorem. I'm finding it very difficult to follow the article about Weierstrass' theorem.
Can someone explain to me what's unjustified about Euler's infinite product representation? Since $\frac{\sin(x)}x$ has a Taylor polynomial representation, and I think all polynomials have roots (in the set of complex numbers), shouldn't it also have a infinite product of roots representation?
Can someone explain to me what Weierstrass' theorem does to justify Euler's representation, and if it's within the ability of a freshman calculus student, can someone show me a proof that is more accessible that the ones I've found by Googling?
Thanks for your time. This is a very interesting problem and very different from the ones I'm used to doing in my calculus class.