First slide

Algebra of complex numbers

Question

If the equation $z^{4} + a_{1} z^{3} + a_{2} z^{2} + a_{3} z + a_{4} = 0$ , where a1, a2, a3, a4 are real coefficients different from zero, has a purely imaginary root, then the expression $\frac{a_{3}}{a_{1} a_{2}} + \frac{a_{1} a_{4}}{a_{2} a_{3}}$ has the value equal to

Moderate

A

0
B

1
C

-2
D

2

Solution

Let xi be the root where x $\neq$ 0 and x $\in$ R
$x^{4} - a_{1} x^{3} i - a_{2} x^{2} + a_{3} xi + a_{4} = 0$
$\Rightarrow x^{4} - a_{2} x^{2} + a_{4} = 0$ (1)
and $a_{1} x^{3} - a_{3} x = 0$ (2)
From Eq. (2),
$a_{1} x^{2} - a_{3} = 0$
$\Rightarrow x^{2} = \frac{a_{3}}{a_{1}} as x \neq 0)$
Putting the value of x² in Eq. (1), we get
$\frac{a_{3}^{2}}{a_{1}^{2}} - \frac{a_{2} a_{3}}{a_{1}} + a_{4} = 0$
$\Rightarrow a_{3}^{2} + a_{4} a_{1}^{2} = a_{1} a_{2} a_{3}$
$\Rightarrow \frac{a_{3}}{a_{1} a_{2}} + \frac{a_{1} a_{4}}{a_{2} a_{3}} = 1 (dividing by a_{1} a_{2} a_{3})$

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