Combine like terms to simplify the expression $z^3+4z^2+z+z^2+6z+1$ .

A) $14z^9$
B) $z^3+13z^2$
C) $z^3+5z^2+7z+1$
D) $13z^9+1$
E) $z^3+12z^2+1$

To simplify the expression, we can combine like terms by adding the coefficients of the terms with the same variable exponent.

Starting with the highest degree term, we have:
$$z^3+\underset{\text{first}}{4z^2}+z+\underset{\text{second}}{z^2}+\underset{\text{third}}{6z}+1$$

Combining like terms, we have:
$$z^3+(4z^2+z^2)+(z+6z)+1=z^3+5z^2+7z+1$$

Therefore, the simplified expression is $z^3+5z^2+7z+1$ , which is answer choice C.