Evaluate the following :
x4 – 4x3 + 4x2 + 8x + 44, when x = 3 + 2i
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)2
⇒ 5
∴ x4-4x3+4x2+8x+44=5