Difference between revisions of "Remainder Theorem"
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Revision as of 12:08, 15 July 2021
Theorem
The Remainder Theorem states that the remainder when the polynomial is divided by
(usually with synthetic division) is equal to the simplified value of
.
Proof
Let , where
is the polynomial,
is the divisor,
is the quotient, and
is the remainder. This equation can be rewritten as
If
, then substituting for
results in
Extension
An extension of the Remainder Theorem could be used to find the remainder of a polynomial when it is divided by a non-linear polynomial. Note that if is a polynomial,
is the quotient,
is a divisor, and
is the remainder, the polynomial can be written as
Note that the degree of
is less than the degree of
. Let
be a root of
, where
is an integer and
. That means for all
,
Thus, the points
are on the graph of the remainder. If all the roots of
are unique, then a system of equations can be made to find the remainder
.
Examples
Introductory
- What is the remainder when
is divided by
?
Solution: Using synthetic or long division we obtain the quotient . In this case the remainder is
. However, we could've figured that out by evaluating
. Remember, we want the divisor in the form of
.
so
.
.