Consider the equation $x^{9} + 5 x^{8} - x^{3} + 7 x + 2 = 0$ . Find the number of imaginary roots.

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Solution

Give equation, $x^{9} + 5 x^{8} - x^{3} + 7 x + 2 = 0$ , has two changes of signs. Hence there can be a maximum of 2 positive roots.

$f (- x) = - x^{9} + 5 x^{8} + x^{3} - 7 x + 2 = 0$ , which has three changes of signs. Hence the given equation has a maximum of 3 negative roots.

Now, as the equation is of $9^{t h}$ degree, it must have at least $(9 - 2 - 3) = 4$ imaginary roots.

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