Difference between revisions of "Remainder Theorem"
Rockmanex3 (talk | contribs) m (→Proof) |
Rockmanex3 (talk | contribs) (Extension of Remainder Theorem) |
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<cmath>p(a) = q(a) \cdot 0 + r(a)</cmath> | <cmath>p(a) = q(a) \cdot 0 + r(a)</cmath> | ||
<cmath>p(a) = r(a)</cmath> | <cmath>p(a) = r(a)</cmath> | ||
+ | |||
+ | ==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 <math>p(x)</math> is a polynomial, <math>q(x)</math> is the quotient, <math>d(x)</math> is a divisor, and <math>r(x)</math> is the remainder, the polynomial can be written as | ||
+ | <cmath>p(x) = q(x)d(x) + r(x)</cmath> | ||
+ | Note that the degree of <math>r(x)</math> is less than the degree of <math>d(x)</math>. Let <math>a_n</math> be a root of <math>d(x)</math>, where <math>n</math> is an integer and <math>1 \le n \le \text{deg } d</math>. That means for all <math>a_n</math>, | ||
+ | <cmath>p(a_n) = r(a_n)</cmath> | ||
+ | Thus, the points <math>(a_n,p(a_n))</math> are on the graph of the remainder. If all the roots of <math>d(x)</math> are unique, then a [[system of equations]] can be made to find the remainder <math>r(x)</math>. | ||
==Examples== | ==Examples== |
Revision as of 01:03, 19 June 2018
Contents
[hide]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
.
.
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