I'm trying to find a way to generalize the expression of polynomials of degree $n-1$ such that $$ k_1+k_2x+k_3x^2+k_4x^3+\dots+k_nx^{n-1}=\frac {k_1}{k_n}+\frac{k_2}{k_n}x+\frac{k_3}{k_n}x^2+\frac{k_4}{k_n}x^3+\dots+x^{n-1} $$ $$ =(\lambda_1+x)(\lambda_2+x)(\lambda_3+x)(\lambda_4+x)\dots(\lambda_{n-1}+x) $$ My goal is to find an expression for $\lambda_i$ in terms of $k_i$, and I've more or less doing this by induction (I think that's the word for it; I'm an engineer, not a mathematician). I started with a polynomial of degree 4 and have come up with the following system of equations: $$ \left( \begin{array}{c} \begin{aligned} \frac{k_4}{k_5}&=\lambda_1+\lambda_2+\lambda_3+\lambda_4 \\ \frac{k_3}{k_5}&=\lambda_1 \lambda_2+\lambda_1 \lambda_3+\lambda_1 \lambda_4+\lambda_2 \lambda_3+\lambda_2 \lambda_4+\lambda_3 \lambda_4\\ \frac{k_2}{k_5}&=\lambda_1 \lambda_2 \lambda_3+\lambda_1 \lambda_2 \lambda_4+\lambda_1 \lambda_3 \lambda_4+\lambda_2 \lambda_3 \lambda_4\\ \frac{k_1}{k_5}&=\lambda_1 \lambda_2 \lambda_3 \lambda_4\\ \end{aligned} \end {array} \right) $$
I theorize that if generalized to a polynomial of degree $n-1$, the system will be:
$$ \left( \begin {array}{c} \begin {aligned} \frac {k_{n-1}}{k_n}&=\sum_{i=1}^{n-1}\lambda_i\\ \frac {k_{n-2}}{k_n}&=\lambda_1 \sum_{i=2}^{n-1} \lambda_i+\lambda_2 \sum_{i=3}^{n-1} \lambda_i+\dots+\prod_{i=n-2}^{n-1} \lambda_i\\ \frac {k_{n-3}}{k_n}&=\lambda_1 \lambda_2 \sum_{i=3}^{n-1}\lambda_i+\lambda_2 \lambda_3 \sum_{i=4}^{n-1}\lambda_i+\dots+\prod_{i=n-4}^{n-3}\sum_{i=n-2}^{n-1}\lambda_i+\lambda_{n-1}\prod_{n-3}^{n-2}\lambda_i \\ &\vdots\\ \frac{k_1}{k_n}&=\prod_{i=1}^{n-1}\lambda_i \end {aligned} \end {array} \right) $$
If anyone knows of any way I could go about solving the system of equations or taking some other approach to the problem, your input would be very much appreciated.