Editing Polynomial remainder theorem

{{Short description|On the remainder of division by x – r}}
{{redirect|Little Bézout's theorem|the intersection number of two algebraic curves|Bézout's theorem|a relation in the theory of greatest common divisors|Bézout's identity}}

In [[algebra]], the '''polynomial remainder theorem''' or '''little Bézout's theorem''' (named after [[Étienne Bézout]])<ref>{{cite journal |author=Piotr Rudnicki |title=Little Bézout Theorem (Factor Theorem) |journal=Formalized Mathematics |volume=12 |issue=1 |year=2004 |pages=49–58 |url=http://mizar.org/fm/2004-12/pdf12-1/uproots.pdf}}</ref> is an application of [[Euclidean division of polynomials]]. It states that, for every number <math>r</math>, any [[polynomial]] <math>f(x)</math> is the sum of <math>f(r)</math> and the product of <math>x-r</math> and a polynomial in <math>x</math> of degree one less than the degree of <math>f</math>. In particular, <math>f(r)</math> is the remainder of the Euclidean division of <math>f(x)</math> by <math>x-r</math>, and <math>x-r</math> is a [[divisor]] of <math>f(x)</math> [[if and only if]] <math>f(r)=0</math>,<ref>Larson, Ron (2014), College Algebra, Cengage Learning</ref> a property known as the [[factor theorem]].

== Examples ==
=== Example 1 ===
Let <math>f(x) = x^3 - 12x^2 - 42</math>. Polynomial division of <math>f(x)</math> by <math>(x-3)</math> gives the quotient <math>x^2 - 9x - 27</math> and the remainder <math>-123</math>. By the polynomial remainder theorem, <math>f(3)=-123</math>.

===Example 2===
Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial <math>f(x) = ax^2 + bx + c</math> by using algebraic manipulation:
<math display="block">\begin{align}
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{align}</math>

So, 
<math display="block">f(x) = (x - r)(ax + ar + b) + f(r), </math>
which is exactly the formula of Euclidean division. 

The generalization of this proof to any degree is given below in {{slink||Direct proof}}.

== Proofs ==

=== Using Euclidean division ===
The polynomial remainder theorem follows from the theorem of [[Euclidean division of polynomials|Euclidean division]], which, given two polynomials {{math|''f''(''x'')}} (the dividend) and {{math|''g''(''x'')}} (the divisor), asserts the existence (and the uniqueness) of a quotient {{math|''Q''(''x'')}} and a remainder {{math|''R''(''x'')}} such that
:<math>f(x)=Q(x)g(x) + R(x)\quad \text{and}\quad R(x) = 0 \ \text{ or } \deg(R)<\deg(g).</math>

If the divisor is <math>g(x) = x-r,</math> where r is a constant, then either {{math|1=''R''(''x'') = 0}} or its degree is zero; in both cases,  {{math|''R''(''x'')}} is a constant that is independent of {{math|''x''}}; that is 
:<math>f(x)=Q(x)(x-r) + R.</math>

Setting <math>x=r</math> in this formula, we obtain:
:<math>f(r)=R.</math>

=== Direct proof ===
A [[constructive proof]]{{mdash}}that does not involve the existence theorem of Euclidean division{{mdash}}uses the identity 
:<math>x^k-r^k=(x-r)(x^{k-1}+x^{k-2}r+\dots+xr^{k-2}+r^{k-1}).</math>
If <math>S_{k}</math> denotes the large factor in the right-hand side of this identity, and 
:<math>f(x)=a_nx^n+a_{n-1}x^{n-1} + \cdots + a_1x +a_0,</math>
one has 
:<math>f(x)-f(r)=(x-r)(a_n S_n +\cdots + a_2S_2 +a_1),</math>
(since <math>S_1=1</math>).

Adding <math>f(r)</math> 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 <math>f(r)</math> by calculating the remainder, <math>R</math>. Although [[polynomial long division]] is more difficult than evaluating the [[Function (mathematics)|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.<ref>Larson, Ron (2011), Precalculus with Limits, Cengage Learning</ref>

== References ==
{{reflist}}
{{Authority control}}
[[Category:Theorems about polynomials]]