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Question
(1+x2)(1+y2)(1+z2) can be expressed as (1−∑xy)2+(∑x−xyz)2. If this is true enter 1, else enter 0.
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Solution
E=[(1+xi)(1+yi)(1+zi)]
[(1−xi)(1−yi)(1−zi)]
=(1+iS1+i2S2+i3S3)(1−iS1+i2S2−i3S3)
=[(1−S2)+i(S1−S3)][(1−S2)−i(S1−S3)]
=(1−S2)2+(S1−S3)2
where S1=∑x,S2=∑xy,S3=xyz
[(1−xi)(1−yi)(1−zi)]
=(1+iS1+i2S2+i3S3)(1−iS1+i2S2−i3S3)
=[(1−S2)+i(S1−S3)][(1−S2)−i(S1−S3)]
=(1−S2)2+(S1−S3)2
where S1=∑x,S2=∑xy,S3=xyz
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