Hexotic Equation - 3
Algebra
Level
4
Find the sum of all non-real roots of the equation
The answer is 1.667.
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1 solution
Moderator note:
Other than finding the factorization, what other ways do we have to obtain the final factorized form?
f ( x ) = 6 x 6 − 2 5 x 5 + 3 1 x 4 − 3 1 x 2 + 2 5 x − 6 = 6 ( x 6 − 1 ) − 2 5 x ( x 4 − 1 ) + 3 1 x 2 ( x 2 − 1 ) = 6 ( x 2 − 1 ) ( x 4 + x 2 + 1 ) − 2 5 x ( x 2 − 1 ) ( x 2 + 1 ) + 3 1 x 2 ( x 2 − 1 ) = ( x 2 − 1 ) ( 6 x 4 − 2 5 x 3 + 3 7 x 2 − 2 5 x + 6 ) = ( x 2 − 1 ) ( 6 x 4 − 1 5 x 3 + 6 x 2 − 1 0 x 3 + 2 5 x 2 − 1 0 x + 6 x 2 − 1 5 x + 6 ) = ( x 2 − 1 ) ( 3 x 2 ( 2 x 2 − 5 x + 2 ) − 5 x ( 2 x 2 − 5 x + 2 ) + 3 ( 2 x 2 − 5 x + 2 ) ) = ( x + 1 ) ( x − 1 ) ( 2 x − 1 ) ( x − 2 ) ( 3 x 2 − 5 x + 3 )
Therefore the real roots of f ( x ) are − 1 , 2 1 , 1 and 2 and the non-real roots come from 3 x 3 − 5 x + 3 = 0 , which means that their sum is 3 5 = 1 . 6 6 7 .