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Evaluate the following :  x4 – 4x3 + 4x2 + 8x + 44, when x = 3 + 2i

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Given: 

⇒ x=3+2i

x4 – 4x3 + 4x2 + 8x + 44

⇒ (3+2i)4-4(3+2i)3+4(3+2i)2+8(3+2i)+44 

⇒ (34+4(3)3(2i)+6(3)2(2i)2+4(3)(2i)3+(2i)4)-4(33+3(3)2(2i)+3(3)(2i)2+(2i)3)+4(32+(2i)2+2(3) (2i))+24+16i+44 

⇒ 81+216i+216i2+96i3+16i4-108-216i-144i2-32i3+36+16i2+48i+24+16i+44 

We know that i2=-1 

⇒ 77+64i+88i2+64i3+16i4

 ⇒ 77+64i+88(-1)+64(-1)(i)+16(-1)

⇒ 5 

∴ x4-4x3+4x2+8x+44=5

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