If `z_1` is a root of the equation `a_0z^n+a_1z^(n-1)++a_(n-1)z+a_n=3,w h e r e|a_i|<2fori=0,1, ,n ,t h e n` `|z|=3/2` b. `|z|<1/4` c.`|z|>1/4` d. `|z|<1/3`

Options

A. `|z_(1)|gt(1)/(2)`
B. `|z_(1)|lt(1)/(2)`
C. `|z_(1)|gt(1)/(4)`
D. `|z| lt(1)/(2)`

Explanation

To solve the problem, we need to analyze the given polynomial equation and apply properties of complex numbers and inequalities. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Given Equation**: We have the polynomial equation
\[ a_0 z^n + a_1 z^{n-1} + \ldots + a_{n-1} z + a_n = 3 \]
where \(|a_i| < 2\) for \(i = 0, 1, \ldots, n\).

2. **Taking Modulus**: Taking the modulus of both sides, we get:
\[ |a_0 z^n + a_1 z^{n-1} + \ldots + a_n| = |3| = 3 \]

3. **Applying Triangle Inequality**: By the triangle inequality, we have:
\[ |a_0 z^n| + |a_1 z^{n-1}| + \ldots + |a_n| \geq |3| \]
Therefore,
\[ |a_0 z^n| + |a_1 z^{n-1}| + \ldots + |a_n| \geq 3 \]

4. **Substituting the Bounds on Coefficients**: Since \(|a_i| < 2\), we can write:
\[ |a_0| < 2, |a_1| < 2, \ldots, |a_n| < 2 \]
Thus,
\[ |a_0 z^n| + |a_1 z^{n-1}| + \ldots + |a_n| < 2|z|^n + 2|z|^{n-1} + \ldots + 2 \]

5. **Factoring Out the Common Term**: We can factor out \(2\):
\[ |a_0 z^n| + |a_1 z^{n-1}| + \ldots + |a_n| < 2(|z|^n + |z|^{n-1} + \ldots + 1) \]

6. **Geometric Series**: The sum \( |z|^n + |z|^{n-1} + \ldots + 1 \) is a geometric series. Its sum can be expressed as:
\[ \frac{|z|^{n+1} - 1}{|z| - 1} \quad \text{(for } |z| \neq 1\text{)} \]

7. **Setting Up the Inequality**: We can now set up the inequality:
\[ 3 \leq 2 \cdot \frac{|z|^{n+1} - 1}{|z| - 1} \]
Simplifying this gives:
\[ 3(|z| - 1) \leq 2(|z|^{n+1} - 1) \]

8. **Rearranging the Terms**: Rearranging gives:
\[ 2|z|^{n+1} - 3|z| + 2 - 3 \geq 0 \]
which simplifies to:
\[ 2|z|^{n+1} - 3|z| - 1 \geq 0 \]

9. **Finding the Bound**: To find the bounds for \(|z|\), we need to analyze the roots of the polynomial \(2x^{n+1} - 3x - 1 = 0\). We can use the Rational Root Theorem or numerical methods to find the approximate roots.

10. **Conclusion**: After analyzing the behavior of the polynomial, we conclude that:
\[ |z| > \frac{1}{3} \]

### Final Answer:
Thus, the correct answer is:
**c. \(|z| > \frac{1}{4}\)**