Let α 1 , α 2 , … , α 7 α 1 , α 2 , … , α 7 be the roots of the equation x 7 + 3 x 5 - 13 x 3 - 15 x = 0 and α 1 ≥ α 2 ≥ … ≥ α 7 . Then, α 1 α 2 - α 3 α 4 + α 5 α 6 is equal to _______

Step-by-step solution

Given: x 7 + 3 x 5 - 13 x 3 - 15 x = 0 ⇒ x x 6 + 3 x 4 - 13 x 2 - 15 = 0 So, x = 0 is one of the root. Now, x 6 + 3 x 4 - 13 x 2 - 15 = 0 Put x 2 = t , then we have t 3 + 3 t 2 - 13 t - 15 = 0 ⇒ t - 3 t 2 + 6 t + 5 = 0 ⇒ t - 3 t + 1 t + 5 = 0 So, t = 3 , t = - 1 , t = - 5 Now we are getting x 2 = 3 , x 2 = - 1 , x 2 = - 5 ⇒ x = ± 3 ,   ± i , ± 5 i From the given condition α 1 ≥ α 2 ≥ … . ≥ α 7 We can clearly say that α 7 = 0 and