# What would the cosets of this subgroup be?

I'm currently studying cosets and the first isomorphism theorem, and I am having trouble understanding what the cosets of this subgroup would be:

We take the group homomorphism $\phi: \mathbb{R}[x] \to \mathbb{R}$ where $\phi$ is the sum of the coefficients of the polynomial. This is clearly a homomorphism because when you add two polynomials, you simply add them term by term. This homomorphism has the kernel of all of the polynomials $p$ in which $p(1)=0$ since $p(1)$ is the sum of the coeffecients.

Since this kernel is a normal subgroup, the set of its cosets must form a group structure, but I'm having trouble seeing what the cosets of this set would be.

If we call this subgroup $K$, then we call a coset: $a+K=\left\{ a+k : k \in K \right\}$ where $a \in \mathbb{R}[x]$.

When I picture doing this with some arbitrary polynomial, I envision that every polynomial can then be constructed, so I don't see how this necesarily has any distinct cosets, yet the first isomorphism theorem tells us that the set of these cosets are isomorphic to $\mathbb{R}$. Can someone help me understand this?

But every polynomial doesn't have $p(1) = 0$.
Rather than writing $\{a + k : k \in K\}$, try writing the equivalent $\{p(x) + a(x): p(1) = 0\}$. Note that we don't have any restriction on $a(1)$.