Question 128933
f(1)=1^3-8(1)^2+17(1)-9=1
f(x)-f(1)=(x^3-8x^2+17x-9)-1=x^3-8x^2+17x-10
let take g(x)= x^3-8x^2+17x-10 so 

1 is a root of g(x)

The Factor Theorem.   x − r  is a factor of a polynomial P(x) if and only if  r is a root of P(x).

so you have to do the division of g(x)/(x-1) 


   1  -8   17  -10
1      1   -7   10
   1  -7   10    0

so g(x)=(x-1)(x^2-7x+10)

now you have to solve x^2-7x+10=0 using formula solutions are (7+sqrt(9))/2 and
(7-sqrt(9))/2 so the other roots are 5 and 2

R: x=2 and x=5