Question

The roots of the polynomial, 𝑓(𝑧)=𝑧 ⁴-8𝑧 ³+27𝑧 ²-38𝑧+26, are 𝑧₁, 𝑧₂, 𝑧₃,& 𝑧₄, where 𝑧 is a complex variable. Which of the following statements is correct?

• A. \(\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= -\frac{4}{19}\)
• B. \(\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= \frac{4}{13}\)
• C. \(\frac{ 𝑧_1𝑧_2𝑧_3𝑧_4 }{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}= -\frac{26}{27}\)
• D. \(\frac{ 𝑧_1𝑧_2𝑧_3𝑧_4 }{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}= -\frac{13}{19}\)

Answer 1

The Correct Option is B

By Vieta’s formulas for a polynomial, the sum and product of the roots of the polynomial are related to the coefficients of the polynomial.

For the given polynomial:

\[ f(z) = z^4 - 8z^3 + 27z^2 - 38z + 26 \]

Step 1: Sum of the Roots

The sum of the roots, \( z_1 + z_2 + z_3 + z_4 \), is equal to the coefficient of \( z^3 \) (with the opposite sign):

\[ z_1 + z_2 + z_3 + z_4 = 8 \]

Step 2: Product of the Roots

The product of the roots, \( z_1 z_2 z_3 z_4 \), is equal to the constant term (with the opposite sign):

\[ z_1 z_2 z_3 z_4 = 26 \]

Step 3: Compute the Required Ratio

The required ratio is:

\[ \frac{z_1 + z_2 + z_3 + z_4}{z_1 z_2 z_3 z_4} = \frac{8}{26} = \frac{4}{13} \]

Conclusion:

Thus, the correct answer is option (B).