Polynomial remainder theorem
In algebra, the polynomial remainder theorem states that the remainder upon dividing any polynomial by a linear polynomial
, both with complex coefficients, is equal to
.
Contents
Proof
We use Euclidean polynomial division with dividend and divisor
. The result states that there exists a quotient
and remainder
such that
with
. We wish to show that
is equal to the constant
. Because
,
. Hence,
is a constant,
. Plugging this into our original equation and rearranging a bit yields
After substituting
into this equation, we deduce that
; thus, the remainder upon diving
by
is equal to
, as desired.
Generalization
The strategy used in the above proof can be generalized to divisors with degree greater than . A more general method, with any dividend
and divisor
, is to write
, and then substitute the zeroes of
to eliminate
and find values of
. Example 2 showcases this strategy.
Examples
Here are some problems with solutions that utilize the remainder theorem and its generalization.
Example 1
What is the remainder when is divided by
?
Solution: Although one could use long or synthetic division, the remainder theorem provides a significantly shorter solution. Note that , and
. A common mistake is to forget to flip the negative sign and assume
, but simplifying the linear equation yields
. Thus, the answer is
, or
, which is equal to
.
.
Example 2
[Insert problem involving the generalization of the remainder theorem]