If the equation z 4 + a 1 z 3 + a 2 z 2 + a 3 z + a 4 = 0 where a 1 , a 2 , a 3 , a 4 are real coefficient different from zero has purely imaginary roots then find the value of the exp

Question

If the equation z 4 + a 1 ​ z 3 + a 2 ​ z 2 + a 3 ​ z + a 4 ​ = 0 where a 1 ​ , a 2 ​ , a 3 ​ , a 4 ​ are real coefficient different from zero has purely imaginary roots then find the value of the expression a 1 ​ a 2 ​ a 3 ​ ​ + a 2 ​ a 3 ​ a 1 ​ a 4 ​ ​ .

TestHub · Accepted Answer

Let z = i y ⇒ y 4 − a 1 ​ y 3 i − a 3 ​ y 2 + i a 3 ​ y + a 4 ​ = 0 ⇒ y 4 − a 2 ​ y 2 + a 4 ​ = 0 − ( 1 ) and − a 1 ​ y 3 + a 3 ​ y = 0 ⇒ y = 0 or y 2 = a 1 ​ a 3 ​ ​ _(2) From (1) and (2) ​ a 1 2 ​ a 3 2 ​ ​ − a 2 ​ a 1 ​ a 3 ​ ​ + a 4 ​ = 0 ​ ⇒ a 1 ​ a 2 ​ a 3 ​ ​ + a 2 ​ a 3 ​ a 1 ​ a 4 ​ ​ = 1 ​

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