"Let $\alpha_1,\alpha_2,....,\alpha_7$ be the roots of the equation ${x^7} + 3{x^5} - 13{x^3} - 15x = 0$ and $|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|$. Then $\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$ is equal to _________."

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Accepted Answer

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${x^7} + 3{x^5} - 13{x^3} - 15x = 0$

$x({x^6} + 3{x^4} - 13{x^2} - 15) = 0$

$x = 0 = {\alpha _7}$

Let ${x^2} = t$

${t^3} + 3{t^2} - 13t - 15 = 0$

$(t + 1)(t + 5)(t - 3) = 0$

$t = {x^2} = - 1, - 5,3$

$x\, = \, \pm \,i, \pm \,\sqrt 5 i, \pm \,\sqrt 3$

$${\alpha _1},{\alpha _2} = \pm \,\sqrt 5 i,{\alpha _3},{\alpha _4} = \pm \,\sqrt 3 ,{\alpha _5},{\alpha _6} = \pm \,i$$

$${\alpha _1}{\alpha _2} - {\alpha _3}{\alpha _4} + {\alpha _5}{\alpha _6} = 5 + 3 + 1 = 9$$

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Let $\alpha_1,\alpha_2,....,\alpha_7$ be the roots of the equation ${x^7} + 3{x^5} - 13{x^3} - 15x = 0$ and $|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|$. Then $\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$ is equal to _________.

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Let $\alpha_1,\alpha_2,....,\alpha_7$ be the roots of the equation ${x^7} + 3{x^5} - 13{x^3} - 15x = 0$ and $|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|$. Then $\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$ is equal to _________.

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