Question

If z = 3 - 4i , then z^4 - 3z^3 + 3z^2 + 99z - 95 is equal to

Answer 1

It is given that, z= 3- 4 i.

Then , $z^{4} - 3z^{3} + 3z^{2} + 99 z - 95$

→$z^{4}$=$(3-4 i )^{4}=(3-4 i )^{2}\times(3-4 i )^{2}=[9-16-24 i]\times[9-16-24 i]=[-7-24 i]^{2}= -527+336 i$, where i² = -1

→(3- 4 i)³= 27+ 64 i -108 i-144= -117 -44 i → [ a-b]³=a³ - b³ - 3 a² b+ 3 ab², where i³= -i and i²= -1

→z²=(3- 4 i)²=9- 16- 24 i= -7- 24 i

→99 (3- 4 i)= 297 - 396 i

Substituting the values in the expression $z^{4} - 3z^{3} + 3z^{2} + 99 z - 95$ =  -527 + 336 i - 3( -117 -44 i) + 3( -7 -24 i) + 297 - 396 i -95

             =  -527 + 336 i + 351 + 132 i - 21 -72 i+ 297 -396 i-95

            = -176+181 +468 i -468 i

            = 5 + 0 i

            =5