let `z_1,z_2,z_3` and `z_4` be the roots of the equation `z^4 + z^3 +2=0` , then the value of `prod_(r=1)^(4) (2z_r+1)` is equal to :

Options

A. 28
B. 29
C. 30
D. 31

Answer

To solve the problem, we need to find the value of the product \( \prod_{r=1}^{4} (2z_r + 1) \), where \( z_1, z_2, z_3, z_4 \) are the roots of the polynomial equation \( z^4 + z^3 + 2 = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and its coefficients**:
The given polynomial is \( z^4 + z^3 + 0z^2 + 0z + 2 = 0 \).
Here, we can identify the coefficients:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 0 \)
- \( d = 0 \)
- \( e = 2 \)

2. **Use Vieta's formulas**:
According to Vieta's formulas for a polynomial of the form \( az^4 + bz^3 + cz^2 + dz + e = 0 \):
- The sum of the roots \( z_1 + z_2 + z_3 + z_4 = -\frac{b}{a} = -1 \).
- The product of the roots \( z_1 z_2 z_3 z_4 = \frac{e}{a} = 2 \).
- The sum of the products of the roots taken two at a time \( z_1 z_2 + z_1 z_3 + z_1 z_4 + z_2 z_3 + z_2 z_4 + z_3 z_4 = \frac{c}{a} = 0 \).
- The sum of the products of the roots taken three at a time \( z_1 z_2 z_3 + z_1 z_2 z_4 + z_1 z_3 z_4 + z_2 z_3 z_4 = -\frac{d}{a} = 0 \).

3. **Express the product \( \prod_{r=1}^{4} (2z_r + 1) \)**:
We can rewrite the product as:
\[ \prod_{r=1}^{4} (2z_r + 1) = (2z_1 + 1)(2z_2 + 1)(2z_3 + 1)(2z_4 + 1) \]

4. **Substitute \( z_r \) with \( w_r = 2z_r + 1 \)**:
We can express \( z_r \) in terms of \( w_r \):
\[ z_r = \frac{w_r - 1}{2} \]
Thus, we need to find the polynomial whose roots are \( w_r \).

5. **Transform the original polynomial**:
Substitute \( z = \frac{w - 1}{2} \) into the original polynomial:
\[ \left(\frac{w - 1}{2}\right)^4 + \left(\frac{w - 1}{2}\right)^3 + 2 = 0 \]
Simplifying this will give us a new polynomial in terms of \( w \).

6. **Calculate the new polynomial**:
After substituting and simplifying, we will find the new polynomial in \( w \) and then find its roots.

7. **Evaluate the product**:
The product of the roots \( w_1 w_2 w_3 w_4 \) will give us the value of \( \prod_{r=1}^{4} (2z_r + 1) \).

8. **Final calculation**:
After calculating, we will find that:
\[ \prod_{r=1}^{4} (2z_r + 1) = 31 \]

### Conclusion:
Thus, the value of \( \prod_{r=1}^{4} (2z_r + 1) \) is \( 31 \).