I need your help on how to show the existence of a bialgebra pairing: for the polynomial ring $K[x]$ and $x \in P(K[x])$ there is bialgebra pairing $t: K[x] \times K[x] \to k$ such that $t(x,x)=1$.

I need your help on how to show the existence of a bialgebra pairing: for the polynomial ring $k[x]$ over a field $k$, there is a bialgebra pairing $t:k[x]\otimes k[x]→k$ such that $t(x,x)=1$. What is the unique bialgebra pairing satisfying $t(x,x)=1$?

I need your help on how to show that existence of bialgebra pairing: Given that for the polynomial ring $K[x]$ and $x \in P(K[x])$ there is bialgebra pairing $\t: K[x] X K[x] \to k$

such that $\\t(x,x)=1$. Thanks. I really need your help.

I need your help on how to show the existence of a bialgebra pairing: for the polynomial ring $K[x]$ and $x \in P(K[x])$ there is bialgebra pairing $t: K[x] \times K[x] \to k$ such that $t(x,x)=1$.