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Simplify $(1-2i)(1-i)(1+i)(1+2i)$.

 Feb 2, 2018
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Simplify the following:

(-2 i + 1) (-i + 1) (i + 1) (2 i + 1)

 

(1 - 2 i) (1 - i) = 1×1 + 1 (-i) + -2 i×1 + -2 i (-i) = 1 - i - 2 i - 2 = -1 - 3 i:

-3 i - 1 (1 + i) (2 i + 1)

 

Factor -1 from -3 i - 1:

-(3 i + 1) (1 + i) (2 i + 1)

 

(1 + 3 i) (1 + i) = 1×1 + 1 i + 3 i×1 + 3 i×i = 1 + i + 3 i - 3 = -2 + 4 i:

-4 i - 2 (2 i + 1)

 

Factor 2 out of 4 i - 2 giving 2 (2 i - 1):

-2 (2 i - 1) (2 i + 1)

 

(-1 + 2 i) (1 + 2 i) = -1 - 2 i + 2 i×1 + 2 i×2 i = -1 - 2 i + 2 i - 4 = -5:

-2×-5

 

-2 (-5) = 10

 Feb 2, 2018

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