For the equation $x^{8} + 6 x^{7} - 5 x^{4} - 3 x^{2} + 2 x - 5 = 0$ , determine the maximum number of real roots possible.

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Solution

The correct option is A 4
$f (x) = x^{8} + 6 x^{7} - 5 x^{4} - 3 x^{2} + 2 x - 5$

Check the number of positive roots= no. of sign changes in f(x) = 3

Check the number of negative roots = no. of sign changes in f(-x)

$f (- x) = x^{8} - 6 x^{7} - 5 x^{4} - 3 x^{2} - 2 x - 5$ . No. of sign changes = 1

Now check for zero as a root. $f (0) = - 5 \neq 0$

So, maximum number of real roots = 3 + 1 = 4. Hence option (a)

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