Q. 184.0( 6 Votes )

Prove that (x + 1 + i) (x + 1 – i) (x – 1 – i) (x – 1 – i) = (x4 + 4).

Answer :

To Prove:

(x + 1 + i) (x + 1 – i) (x – 1 + i) (x – 1 – i) = (x4 + 4)


Taking LHS


(x + 1 + i) (x + 1 – i) (x – 1 + i) (x – 1 – i)


= [(x + 1) + i][(x + 1) – i][(x – 1) + i][(x – 1) – i]


Using (a – b)(a + b) = a2 – b2



= [(x + 1)2 – (i)2] [(x – 1)2 – (i)2]


= [x2 + 1 + 2x – i2](x2 + 1 – 2x – i2]


= [x2 + 1 + 2x – (-1)](x2 + 1 – 2x – (-1)] [ i2 = -1]


= [x2 + 2 + 2x][x2 + 2 – 2x]


Again, using (a – b)(a + b) = a2 – b2


Now, a = x2 + 2 and b = 2x


= [(x2 + 2)2 – (2x)2]


= [x4 + 4 + 2(x2)(2) – 4x2] [(a + b)2 = a2 + b2 + 2ab]


= [x4 + 4 + 4x2 – 4x2]


= x4 + 4


= RHS


LHS = RHS


Hence Proved


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