In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an
application of Euclidean division of polynomials. It states that the remainder of the
division of a polynomial
particular,
by a linear polynomial
is a divisor of
is equal to
In
if and only if
Contents
Examples
Example 1
Let
. Polynomial division of
quotient
and the remainder
by
gives the
. Therefore,
.
Example 2[
Show that the polynomial remainder theorem holds for an arbitrary second degree
polynomial
by using algebraic manipulation:
Multiplying both sides by (x − r) gives
.
Since
is our remainder, we have indeed shown
that
.
Proof
The polynomial remainder theorem follows from the definition of polynomial long
division; denoting the divisor, quotient and remainder by, respectively,
,
, and
, polynomial long division gives a solution of the equation
where the degree of
is less than that of
If we take
as the divisor, giving the degree of
i.e.
.
as 0,
:
Setting
we obtain:
Applications ]
The polynomial remainder theorem may be used to evaluate
by
calculating the remainder, . Although polynomial long division is
more difficult than evaluating the function itself, synthetic division is
computationally easier. Thus, the function may be more "cheaply"
evaluated using synthetic division and the polynomial remainder
theorem.
The factor theorem is another application of the remainder theorem: if
the remainder is zero, then the linear divisor is a factor. Repeated
application of the factor theorem may be used to factorize the
polynomial.
References
1. ^ Piotr Rudnicki (2004). "Little Bézout Theorem (Factor
Theorem)". Formalized Mathematics 12 (1): 49–58.
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Categories:
Polynomials
Theorems in algebra