I am trying to understand the logic behind solving this problem from youtube.
Given the equation:
$p_n-e_1p_{n-1}+\dots+n(-1)^ne_n = 0$
And the Newton-Girard identity:
$ke_k = e_{k-1}p_1-e_{k-2}p_2+\dots+(-1)^{k-1}e_0p_k$
How are these 2 equations related ?
Does it mean $ke_k = 0$ ? That makes no sense, yet looking at the right hand side of both the equations they both look the same.
What is going on? I cant seem to understand.
Consider specific values. Can you go between
$$ p_5-e_1p_4+e_2p_3-e_3p_2+e_4p_1-5e_5=0 $$
and
$$ 5e_5=e_4p_1-e_3p_2+e_2p_3-e_1p_4+p_4? $$
Sure you can; just solve for $5e_5$ by adding it to both sides (and reordering the terms on the one side). If instead $k$ is even then you could subtract $ke_k$ and multiply by $-1$.
(Note $e_0=1$ is the sum of the empty product and if $k=n$ then $p_n=n$ so the identity would have the even more symmetric form $\sum_{i=0}^n (-1)^ie_ip_{n-i}=0$.)