Synthetic substitution is the evaluation of a polynomial at a specified value using the coefficient operations of synthetic division. For
 |
the value 
can be computed by setting 
and successively forming
 |
for 
,
,
..., 0. The last result is 
. This is Horner's rule
written as a synthetic division table, with the polynomial
remainder theorem identifying the final value as the remainder on division by
.
