Editing Polynomial remainder theorem (section)

== 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}}.