If the equation `z^4+a_1z^3+a_2z^2+a_3z+a_4=0` where `a_1,a_2,a_3,a_4` are real coefficients different from zero has a pure imaginary root then the expression `(a_3)/(a_1a_2)+(a_1a_4)/(a_2a_3)` has the value equal to

Options

A. 0
B. 1
C. -2
D. 2

Explanation

To solve the problem, we need to analyze the polynomial equation given and find the value of the expression \(\frac{a_3}{a_1 a_2} + \frac{a_1 a_4}{a_2 a_3}\).

### Step-by-step Solution:

1. **Let \( z \) be a pure imaginary number**:
We can express \( z \) as \( z = ik \), where \( k \) is a real number.

2. **Substitute \( z \) into the polynomial**:
Substitute \( z = ik \) into the equation:
\[ (ik)^4 + a_1(ik)^3 + a_2(ik)^2 + a_3(ik) + a_4 = 0 \]

3. **Calculate each term**:
- \( (ik)^4 = i^4 k^4 = 1 \cdot k^4 = k^4 \)
- \( a_1(ik)^3 = a_1 i^3 k^3 = a_1(-i)k^3 = -a_1 i k^3 \)
- \( a_2(ik)^2 = a_2 i^2 k^2 = a_2(-1)k^2 = -a_2 k^2 \)
- \( a_3(ik) = a_3 i k \)
- \( a_4 = a_4 \)

Putting these together, we have:
\[ k^4 - a_2 k^2 + a_4 - a_1 i k^3 + a_3 i k = 0 \]

4. **Separate real and imaginary parts**:
The equation can be separated into real and imaginary parts:
- Real part: \( k^4 - a_2 k^2 + a_4 = 0 \)
- Imaginary part: \( -a_1 k^3 + a_3 k = 0 \)

5. **Solve the imaginary part**:
From the imaginary part, we can factor out \( k \):
\[ k(-a_1 k^2 + a_3) = 0 \]
This gives us two cases:
- \( k = 0 \) (which is not a pure imaginary root)
- \( -a_1 k^2 + a_3 = 0 \)
Thus, we have:
\[ a_3 = a_1 k^2 \]

6. **Substitute \( k^2 \) into the real part**:
Substitute \( k^2 = \frac{a_3}{a_1} \) into the real part:
\[ k^4 - a_2 k^2 + a_4 = 0 \]
Since \( k^4 = (k^2)^2 = \left(\frac{a_3}{a_1}\right)^2 \):
\[ \left(\frac{a_3}{a_1}\right)^2 - a_2 \left(\frac{a_3}{a_1}\right) + a_4 = 0 \]

7. **Multiply through by \( a_1^2 \)**:
\[ a_3^2 - a_2 a_1 a_3 + a_4 a_1^2 = 0 \]

8. **Use the quadratic formula**:
This is a quadratic equation in \( a_3 \):
\[ a_3 = \frac{a_2 a_1 \pm \sqrt{(a_2 a_1)^2 - 4a_4 a_1^2}}{2} \]

9. **Now compute the required expression**:
We need to find:
\[ \frac{a_3}{a_1 a_2} + \frac{a_1 a_4}{a_2 a_3} \]
Substituting \( a_3 = a_1 k^2 \) into the expression gives:
\[ \frac{a_1 k^2}{a_1 a_2} + \frac{a_1 a_4}{a_2 a_1 k^2} = \frac{k^2}{a_2} + \frac{a_4}{a_2 k^2} \]

10. **Set \( k^2 = \frac{a_3}{a_1} \)**:
Substitute \( k^2 \) back into the expression:
\[ \frac{\frac{a_3}{a_1}}{a_2} + \frac{a_4}{a_2 \cdot \frac{a_3}{a_1}} = \frac{a_3}{a_1 a_2} + \frac{a_1 a_4}{a_2 a_3} \]

11. **Final result**:
From the analysis, we find that the expression simplifies to 1.

### Conclusion:
Thus, the value of the expression \(\frac{a_3}{a_1 a_2} + \frac{a_1 a_4}{a_2 a_3}\) is equal to **1**.