Suppose $f(z)$ and $g(z)$ are polynomials in $z$, and the degree of $g(z)$ is less than the degree of $f(z)$.  If the degree of $f(z)$ is two, what is the degree of $f(z)+g(z)$?
We have $f(z)=a_2 \cdot z^2+a_1 \cdot z+a_0$, and $g(z)=b_1 \cdot z+b_0$, where $a_2$ is nonzero.  Then $f(z)+g(z)=a_2 \cdot z^2+(a_1+b_1) \cdot z+(a_0+b_0)$.  The degree of this polynomial is $\boxed{2}$.
2
Level 3
Algebra
Problem: Suppose $f(z)$ and $g(z)$ are polynomials in $z$, and the degree of $g(z)$ is less than the degree of $f(z)$.  If the degree of $f(z)$ is two, what is the degree of $f(z)+g(z)$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the degree of the sum of two polynomials, we need to understand the properties of polynomial degrees. The degree of a polynomial is defined as the highest power of the variable in the polynomial. 

Given that $f(z)$ is a polynomial of degree 2, we can express it in the general form as $f(z) = a_2 z^2 + a_1 z + a_0$, where $a_2 \neq 0$. 

The polynomial $g(z)$ has a degree less than that of $f(z)$. Therefore, the degree of $g(z)$ can be either 1 or 0. We can express $g(z)$ in the general form as $g(z) = b_1 z + b_0$ (if degree 1) or $g(z) = b_0$ (if degree 0), where $b_1$ or $b_0$ may be zero.

When we add $f(z)$ and $g(z)$, we consider the highest degree terms from both polynomials. The highest degree term in $f(z)$ is $a_2 z^2$, which has a degree of 2. The highest degree term in $g(z)$, whether it is $b_1 z$ or $b_0$, will have a degree of at most 1.

In the case where $g(z)$ has a degree of 1, the highest degree term from $g(z)$ is $b_1 z$, which does not affect the highest degree term from $f(z)$. Thus, the degree of the sum $f(z) + g(z)$ remains determined by the degree of $f(z)$, which is 2.

In the case where $g(z)$ has a degree of 0, the highest degree term from $g(z)$ is simply a constant $b_0$. Again, this does not affect the highest degree term from $f(z)$, and the degree of the sum $f(z) + g(z)$ remains 2.

Since in both scenarios the highest degree term comes from $f(z)$, we conclude that the degree of the polynomial $f(z) + g(z)$ is 2. 

Thus, the final answer is 2.<|END_OF_LATENT|> The final answer is $2$.