If `alpha` is a root of `z^5 +z^3 +z+3=0`, then

Options

A. `|alpha|ge 1`
B. `|alpha|lt 1`
C. `alpha` lies on or out side the unit circle
D. `alpha` lies inside the unit circle `|z|=1`

Answer

To solve the problem, we need to analyze the polynomial equation given and determine the properties of its roots. The equation is:

\[ z^5 + z^3 + z + 3 = 0 \]

We are tasked with finding information about the root \( \alpha \).

### Step 1: Substitute \( z = \alpha \)
Since \( \alpha \) is a root of the polynomial, we can substitute \( z \) with \( \alpha \):

\[ \alpha^5 + \alpha^3 + \alpha + 3 = 0 \]

### Step 2: Rearranging the equation
Rearranging the equation gives us:

\[ \alpha^5 + \alpha^3 + \alpha = -3 \]

### Step 3: Taking the modulus
Next, we take the modulus of both sides of the equation:

\[ |\alpha^5 + \alpha^3 + \alpha| = |-3| \]

This simplifies to:

\[ |\alpha^5 + \alpha^3 + \alpha| = 3 \]

### Step 4: Applying the triangle inequality
Using the triangle inequality, we know that:

\[ |\alpha^5 + \alpha^3 + \alpha| \leq |\alpha^5| + |\alpha^3| + |\alpha| \]

### Step 5: Analyzing the terms
If we assume \( |\alpha| < 1 \), then:

- \( |\alpha^5| < 1 \)
- \( |\alpha^3| < 1 \)
- \( |\alpha| < 1 \)

Thus, we would have:

\[ |\alpha^5| + |\alpha^3| + |\alpha| < 1 + 1 + 1 = 3 \]

This leads to a contradiction since we established that \( |\alpha^5 + \alpha^3 + \alpha| = 3 \). Therefore, our assumption that \( |\alpha| < 1 \) must be incorrect.

### Step 6: Conclusion about the modulus
Since our assumption was wrong, we conclude that:

\[ |\alpha| \geq 1 \]

This means that the modulus of \( \alpha \) is either equal to 1 or greater than 1.

### Step 7: Finding specific values
To further analyze, we can check specific values. For example, if we let \( z = -1 \):

\[ (-1)^5 + (-1)^3 + (-1) + 3 = -1 - 1 - 1 + 3 = 0 \]

This shows that \( z = -1 \) is indeed a root of the polynomial, confirming that \( |\alpha| = 1 \) is possible.

### Final Answer
Thus, we conclude that:

- The modulus of \( \alpha \) is \( |\alpha| \geq 1 \).