Add the following expressions:
`3z^(3)-z^(2)+5, 1-2z+z^(2), 3+2z-z^(3)`

Answer

To add the given algebraic expressions \(3z^3 - z^2 + 5\), \(1 - 2z + z^2\), and \(3 + 2z - z^3\), we will follow these steps:

### Step 1: Write down the expressions
We have the following three expressions:
1. \(3z^3 - z^2 + 5\)
2. \(1 - 2z + z^2\)
3. \(3 + 2z - z^3\)

### Step 2: Combine the expressions
Now, we will combine all three expressions together:
\[ (3z^3 - z^2 + 5) + (1 - 2z + z^2) + (3 + 2z - z^3) \]

### Step 3: Rearrange and group like terms
Next, we will rearrange the expression to group the like terms together:
\[ (3z^3 - z^3) + (-z^2 + z^2) + (-2z + 2z) + (5 + 1 + 3) \]

### Step 4: Simplify each group
Now, we will simplify each group:
- For the \(z^3\) terms: \(3z^3 - z^3 = 2z^3\)
- For the \(z^2\) terms: \(-z^2 + z^2 = 0\)
- For the \(z\) terms: \(-2z + 2z = 0\)
- For the constant terms: \(5 + 1 + 3 = 9\)

### Step 5: Write the final expression
Putting it all together, we get:
\[ 2z^3 + 0 + 0 + 9 = 2z^3 + 9 \]

### Final Answer
The sum of the expressions is:
\[ \boxed{2z^3 + 9} \]