If `z=e^((2pi i)/5)`, then `1+z+z^(2)+z^(3)+5z^(4)+4z^(5)+4z^(6)+4z^(7)+4z^(8)+5z^(9)=`

Options

A. `0`
B. `4z^(3)`
C. `5z^(4)`
D. `-4z^(2)`

Explanation

To solve the expression \(1 + z + z^2 + z^3 + 5z^4 + 4z^5 + 4z^6 + 4z^7 + 4z^8 + 5z^9\) where \(z = e^{\frac{2\pi i}{5}}\), we can follow these steps:

### Step 1: Identify the properties of \(z\)
Since \(z = e^{\frac{2\pi i}{5}}\), we know that \(z^5 = 1\). This means \(z\) is a fifth root of unity.

**Hint:** Recall that the \(n\)th roots of unity satisfy \(z^n = 1\).

### Step 2: Simplify the expression using \(z^5 = 1\)
We can reduce the powers of \(z\) greater than 4 using the property \(z^5 = 1\):
- \(z^5 = 1\)
- \(z^6 = z\)
- \(z^7 = z^2\)
- \(z^8 = z^3\)
- \(z^9 = z^4\)

Substituting these back into the expression gives:
\[ 1 + z + z^2 + z^3 + 5z^4 + 4(1) + 4z + 4z^2 + 4z^3 + 5z^4 \]

### Step 3: Combine like terms
Now, we can combine the terms:
\[ 1 + 4 + (z + 4z) + (z^2 + 4z^2) + (z^3 + 4z^3) + (5z^4 + 5z^4) \]
This simplifies to:
\[ 5 + 5z + 5z^2 + 5z^3 + 10z^4 \]

### Step 4: Factor out the common term
We can factor out the 5:
\[ 5(1 + z + z^2 + z^3 + 2z^4) \]

### Step 5: Use the property of roots of unity
We know that:
\[ 1 + z + z^2 + z^3 + z^4 = 0 \]
Thus, we can rewrite:
\[ 1 + z + z^2 + z^3 + 2z^4 = 2z^4 \]
So, we have:
\[ 5(2z^4) = 10z^4 \]

### Step 6: Substitute \(z^4\)
Since \(z^4 = e^{\frac{8\pi i}{5}}\), we can leave it in this form or calculate its value if needed.

### Final Answer
Thus, the final result is:
\[ 10z^4 \]

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