Answer:
1. -16; 2. +64; 3. 16
Step-by-step explanation:
The formula for the volume of a cylinder is
V = πr²x
Data:
 r = (x - 8)   mm
V = 1024π mm³
Calculations:
1. Find the cubic equation  
V = πr²h
1024π = π(x - 8)² × x
Divide each side by π
1024 = x(x - 8)²  
1024 = x(x² - 16x  + 64)
1024 =    x³ - 16x² + 64x
x³ - 16x² + 64x - 1024 = 0
2. Solve the cubic equation
The general formula for a third-degree polynomial is
f(x) = ax³ + bx² + cx + d
Your polynomial is  
f(x) = x³ - 16x² + 64x - 1024
a = 1; d = -1024
According to the <em>rational roots theorem</em>, the possible roots are
factors of d/factors of a
Factors of d = ±1, ±2, ±4, ±8, ±16, ± 32, ± 64, ±128, ±256, ±512, ±1024
Factors of a = ±1
Potential roots are x = ±1, ±2, ±4, ±8, ±16, ± 32, ± 64, ±128, ±256, ±512, ±1024
That's a lot of possibilities to check by trial and error. I will just use the one that works.
Try x = 16 by synthetic division.
16|1  -16   64  -1024
   <u>|     16     0   1024
</u>
    1     0   64        0
So,
(x³ - 16x² + 64x - 1024)/(x – 16) = x² + 64
and
(x - 16)(x² + 64) = 0
x - 16 = 0        x² + 64 =    0
      x = 16        x²         = -64
                        x          =  ±8i
There is only one real root: x = 16 mm