Roots of Complex Numbers
If α , α 2 , α 3 and α 4 are the non-real roots of x 5 − 1 = 0 , find the value of ( 1 − α ) ( 1 − α 2 ) ( 1 − α 3 ) ( 1 − α 4 ) .
The answer is 5.
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1 solution
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Since the degree of x 5 − 1 is 5 , it must have 5 roots. In particular, x = 1 is the only real root. We can factorise it like this: x 5 − 1 = ( x − 1 ) ( x 4 + x 3 + x 2 + x + 1 ) However, as 1 , α , α 2 , α 3 , α 4 are the 5 roots of x 5 − 1 , by Factor Theorem, we can also write x 5 − 1 = ( x − 1 ) ( x − α ) ( x − α 2 ) ( x − α 3 ) ( x − α 4 ) Since both equations are identities, we must have ( x − α ) ( x − α 2 ) ( x − α 3 ) ( x − α 4 ) = x 4 + x 3 + x 2 + x + 1 Substituting x = 1 , ( 1 − α ) ( 1 − α 2 ) ( 1 − α 3 ) ( 1 − α 4 ) = 1 4 + 1 3 + 1 2 + 1 + 1 = 5
x 5 − 1 = ( x − 1 ) ( x 4 + x 3 + x 2 + x + 1 )
The non-real roots are also roots of x 4 + x 3 + x 2 + x + 1 = 0 , so x 4 + x 3 + x 2 + x + 1 = ( x − α ) ( x − α 2 ) ( x − α 3 ) ( x − α 4 ) .
Hence plug in x = 1 to get the answer as 5 .