Decomposition of a polynomial

Question

My question : Given a polynomial $$ \sum_{n=0}^{N}a_{n}x^{n} $$ cannot be solved in general, but depending on the coefficients, can we know if it can be decomposed as a product of smaller order polynomials of degree 1 ,2 ,3 and 4 so in this case can be factorized and solved?

Answer 1

By the fundamental theorem of algebra we know that every non-zero $n^\text{th}$ degree polynomial with constant complex coefficients has, counted with their algebraic multiplicity, exactly $n$ roots.

So by this means we can decompose, in the complex field, every polynomial into a product of polynomials with degree less than $n$. As an example take a $5^\text{th}$ degree polynomial $P(x)$ with three roots $x_1, x_2, x_3$ such that $$\text{a.m.}(x_1) = 1 \;\; \text{a.m.}(x_2) = 2\;\; \text{a.m.}(x_3) = 2$$ were $\text{a.m.}(x_n)$ is the algebraic multiplicity of $x_n$, then we can write $$P(x) = (x-x_1)(x-x_2)^2(x-x_3)^2$$

Depending on the coefficients you can say something about the roots of the polynomial by using Decartes' rule of signs or with the Routh-Hurwitz critetion. I don't think that, in a general case, you can say anything more than this, but I stand to be corrected. Surely we can say something more by knowing the form of the polynomial