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{{distinguish|Bézout's theorem}}


{{one source|date=March 2012}}
In [[algebra]], the '''polynomial remainder theorem''' or '''little Bézout's theorem'''<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 the remainder of the division of a [[polynomial]] <math>f(x)</math> by a [[linear polynomial]] <math>x-a</math> is equal to <math>f(a) .</math> In particular, <math>x-a</math> is a [[divisor]] of <math>f(x)</math> if and only if <math>f(a)=0.</math>


== Examples ==
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===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>. Therefore, <math>f(3)=-123\,</math>.
 
===Example 2===
Show 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>
\begin{align}
\frac{f(x)}{{x - r}} &= \frac{{a{x^2} + bx + c}}{{x - r}} \\
&= \frac{{ax(x - r) + (b + ar)x + c}}{{x - r}} \\
&= ax + \frac{{(b + ar)(x - r) + c + r(b + ar)}}{{x - r}} \\
&= ax + b + ar + \frac{{c + r(b + ar)}}{{x - r}} \\
&= ax + b + ar + \frac{{a{r^2} + br + c}}{{x - r}}
\end{align}</math>
 
Multiplying both sides by (''x''&nbsp;&minus;&nbsp;''r'') gives
:<math>f(x) = ax^2 + bx + c = (ax + b + ar)(x - r) + {a{r^2} + br + c}</math>.
 
Since <math>R = ar^2 + br + c</math> is our remainder, we have indeed shown that <math>f(r) = R</math>.
 
== Proof ==
 
The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, <math>g(x)\,</math>, <math>q(x)\,</math>, and <math>r(x)\,</math>, polynomial long division gives a solution of the equation
:<math>f(x)=q(x)g(x) + r(x)\,,</math>
where the [[Degree of a polynomial|degree]] of <math>r(x)\,</math> is less than that of <math>g(x)\,</math>.
 
If we take <math>g(x) = x-a\,</math> as the divisor, giving the degree of <math>r(x)\,</math> as 0, i.e. <math>r(x) = r\,</math>:
:<math>f(x)=q(x)(x-a) + r\,.</math>
 
Setting <math>x=a \!\,</math> we obtain:
:<math>f(a)=r\,.</math>
 
== Applications ==
 
The polynomial remainder theorem may be used to evaluate <math>f(a)\,</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.
 
== References ==
{{reflist}}
 
[[Category:Polynomials]]
[[Category:Theorems in algebra]]

Latest revision as of 21:59, 1 September 2014


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