I have trouble understanding Hatcher's definition about attaching $e^{n+1}$ to $S^1 \vee S^n$ via a map $f$ corresponding to a polynomial $p(t)\in \mathbb Z [t,t^{-1}]$ in the exercise 4.2.10. Let's consider the following concrete polynomial $p(t)=2t^3-4(t^{-1})^7+8$. Can someone tell me how this polynomial corresponds to this attaching map specifically and why the homotopy group of the resulting space is $\pi_n(X)=Z[t,t^{-1}]/(p(t))$?
Someone told me that either the coefficients $2,-4, 8$ or the power index $3,7$ should correspond to the degree of $f$, but I didn't understand why the degree theory should be involved here(I do know $\pi_n(S^1 \vee S^n)=\mathbb Z [t,t^{-1}]$, but degree theory is in homology theory, so I can't see the connection between them).
It's a nice but, perhaps, difficult to grasp initially, description of the $n$th homotopy group. First, notice that the universal cover of $X = S^1\vee S^n$ is $\mathbb R$ with a copy of $S^n$ attached at each integer point, call that space $\tilde X$. By collapsing the contractible subspace $\mathbb R \subset \tilde X$, you see that $\tilde X \simeq \bigvee S^n$, where there is one copy of $S^n$ for each integer. Thus, $\pi_n(X) = \pi_n (\tilde X) = \bigoplus_{k\in \mathbb Z} \mathbb Z$. Now if you realize that the action of $\pi_1(X) = \mathbb Z$ on $\pi_n(X)$ is given by the deck transformations on $\tilde X$, then if you label the copies of $\mathbb Z$ using integer powers of $t$, you see why $\pi_n(X) = \mathbb Z[t,t^{-1}]$, the action of $\pi_1$ is by $t$ and the group operation is addition. Now, if you attach a cell along a "polynomial" $p(t) \in \pi_n(X)$ you see that the resulting space $Y$ has universal cover given by attaching $t^k p(t) \in \pi_n(\tilde X)$ to the cover $\tilde X$, call this cover $\tilde Y$. Now, it should be easy, with the right tools (Hurewicz theorem), to see that $\pi_n(Y) = \pi_n(\tilde Y) = \mathbb Z[t,t^{-1}]/(p(t))$.