On the remainder of division by x – r
"Little Bézout's theorem" redirects here. For the intersection number of two algebraic curves, see
Bézout's theorem. For a relation in the theory of greatest common divisors, see
Bézout's identity.
In
algebra, the polynomial remainder theorem or little Bézout's theorem (named after
Étienne Bézout)
[1] is an application of
Euclidean division of polynomials. It states that, for every number
any
polynomial
is the sum of
and the product by
of a polynomial in
of degree less than the degree of
In particular,
is the remainder of the Euclidean division of
by
and
is a
divisor of
if and only if
[2] a property known as the
factor theorem.
Examples
Example 1
Let
. Polynomial division of
by
gives the quotient
and the remainder
. Therefore,
.
Example 2
Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial
by using algebraic manipulation:
![{\displaystyle {\begin{aligned}f(x)-f(r)&=ax^{2}+bx+c-(ar^{2}+br+c)\\&=a(x^{2}-r^{2})+b(x-r)\\&=a(x-r)(x+r)+b(x-r)\\&=(x-r)(ax+ar+b)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21fdb1b910dbee77afcd61778e379f2cd991f09d)
So,
![{\displaystyle f(x)=(x-r)(ax+ar+b)+f(r),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/258b79261bdeade7eb2160b00127de185b873240)
which is exactly the formula of Euclidean division.
The generalization of this proof to any degree is given below in
§ Direct proof.
Proofs
Using Euclidean division
The polynomial remainder theorem follows from the theorem of
Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that
![{\displaystyle f(x)=Q(x)g(x)+R(x)\quad {\text{and}}\quad R(x)=0\ {\text{ or }}\deg(R)<\deg(g).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e88d0a05670afa6add3ae7668717fa7c95f0214)
If the divisor is
where r is a constant, then either R(x) = 0 or its degree is zero; in both cases, R(x) is a constant that is independent of x; that is
![{\displaystyle f(x)=Q(x)(x-r)+R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0e2a518b4b21892c1b022515b2f283894ed015)
Setting
in this formula, we obtain:
![{\displaystyle f(r)=R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1b14164d0e09b91bca8b3f03c5ef34f384365b)
Direct proof
A
constructive proof—that does not involve the existence theorem of Euclidean division—uses the identity
![{\displaystyle x^{k}-r^{k}=(x-r)(x^{k-1}+x^{k-2}r+\dots +xr^{k-2}+r^{k-1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466bd978688fa5022a3b4763691382f733cde88c)
If
denotes the large factor in the right-hand side of this identity, and
![{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8bf90f0688877d87584d33c5368de20afc0a0df)
one has
![{\displaystyle f(x)-f(r)=(x-r)(a_{n}S_{n}+\cdots +a_{2}S_{2}+a_{1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db4e170c376b32f19b98ba1e1b78dd32ac5e1068)
(since
).
Adding
to both sides of this equation, one gets simultaneously the polynomial remainder theorem and the existence part of the theorem of Euclidean division for this specific case.
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.
[3]
References
-
^ Piotr Rudnicki (2004).
"Little Bézout Theorem (Factor Theorem)" (PDF). Formalized Mathematics. 12 (1): 49–58.
-
^ Larson, Ron (2014), College Algebra, Cengage Learning
-
^ Larson, Ron (2011), Precalculus with Limits, Cengage Learning