find all the zeros of x^3-10x^2+48x-64 given that 4-4i is a zero.

Answer 1

The possible rational solutions are {+-1, +-2, +-4, +-8, +-16, +-32, +-64}

2 is a solution, since it makes the polynomial zero when x=2.

Since 4 - 4i is GIVEN to be a solution, then 4 + 4i must also be a solution.

Nevertheless, here is the synthetic division to verify:

2 | 1 -10 48 -64

2 -16 64

_______________

1 -8 32 0

x^2 - 8x + 32 = 0

[8 +or- sqrt( 64 - 4*32) ]/ 2

[8 +or- sqrt( 64 - 128)]/2

[ 8 +or- sqrt(-64)]/2

[ 8 +or- 8i]/2

4 +or- 4i