Let denote a non-real complex number where . Define and .
If and are roots of the equation for constants and , what is the value of ?
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By Vieta's formulas, we know that p = − ( a + b ) and q = a b , so:
p = − ( w + w 2 + w 4 + w 3 + w 5 + w 6 ) = 1 q = ( w + w 2 + w 4 ) ( w 3 + w 5 + w 6 ) = w 4 + w 6 + 1 + w 5 + 1 + w + 1 + w 2 + w 3 = 2
So, 1 0 p + q = 1 0 ( 1 ) + 2 = 1 2