Factor:

x³ - 8x² + 11x + 20

If there are any rational zeros they must be ± divisors
of 20, which are ±1, ±2, ±4, ±5, ±10, ±20.

We try the easiest, 1, using synthetic division:

1| 1 -8  11  20
 |    1  -7   4  
   1 -7   4  24

No that gives remainder 24, not 0

We try the next easiest, -1:

-1| 1 -8  11  20
  |   -1   9 -20   
    1 -9  20   0

That gives remainder 0, so (x + 1) is a factor of 
the polynomial. Therefore

(x + 1)(x² - 9x + 20) 
 
is a partial factorization, but not a complete
factorization, because the binomial in the second
parentheses can be factored, and the final
factorization is:

(x + 1)(x - 5)(x - 4)

Edwin