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Physics Question on Mathematical Methods

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

𝑧1+𝑧2+𝑧3+𝑧4𝑧1𝑧2𝑧3𝑧4=−419\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= -\frac{4}{19}

B

𝑧1+𝑧2+𝑧3+𝑧4𝑧1𝑧2𝑧3𝑧4=−413\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= -\frac{4}{13}

C

𝑧1𝑧2𝑧3𝑧4𝑧1+𝑧2+𝑧3+𝑧4=−2627\frac{ 𝑧_1𝑧_2𝑧_3𝑧_4 }{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}= -\frac{26}{27}

D

𝑧1𝑧2𝑧3𝑧4𝑧1+𝑧2+𝑧3+𝑧4=−1319\frac{ 𝑧_1𝑧_2𝑧_3𝑧_4 }{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}= -\frac{13}{19}

Answer

𝑧1+𝑧2+𝑧3+𝑧4𝑧1𝑧2𝑧3𝑧4=−413\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= -\frac{4}{13}

Explanation

Solution

The correct option is (B): 𝑧1+𝑧2+𝑧3+𝑧4𝑧1𝑧2𝑧3𝑧4=−413\frac{ 𝑧_1 + 𝑧_2 + 𝑧_3 + 𝑧_4}{ 𝑧_1𝑧_2𝑧_3𝑧_4 }= -\frac{4}{13}