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The sum of the real roots of the equation x 5 - 5 x 4 + 9 x 2 - 9 x 2 + 5 x - 1 = 0 is

Correct answer

4

Step-by-step solution

The sum of the coefficients is zero ⇒ x = 1 is a root ⇒ x - 1 is a factor. x 5 - 5 x 4 + 9 x 3 - 9 x 2 + 5 x - 1 = x - 1 x 4 - 4 x 3 + 5 x 2 - 4 x + 1 Now, x 4 - 4 x 3 + 5 x 2 - 4 x + 1 = 0 ⇒ x 2 - 4 x + 5 - 4 x + 1 x 2 = 0 x + 1 x 2 - 4 x + 1 x + 3 = 0 ⇒ x + 1 x = 1 , 3 x + 1 x = 1 ⇒ x 2 - x + 1 = 0 ⇒ x = 1 ± i 3 2 x + 1 x = 3 ⇒ x 2 - 3 x + 1 = 0 ⇒ x = 3 + 5 2 Roots are 1 , 1 ± i 3 2 , 3 ± 5 2 Sum of real roots = 4

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